University of Innsbruck

Master Thesis

Stable Reference Cavityfor Er and Dy MOT Light

Alexander Patscheider

Supervised by Prof. Dr. Francesca FerlainoInstitute of Experimental Physics, University of Innsbruck

Submitted to the Faculty of

Mathematics, Computer Science and Physics

February 10, 2017

2

AbstractIn this thesis we describe the setup and characterization of a reference cavity for thestabilization of laser light at several wavelengths. On the one hand, the cavity is used tostabilize light used for the magneto-optical trap of erbium and dysprosium atoms. On theother hand, we want to stabilize laser light for Rydberg excitation at a future date of theexperiment. Additionally, the thesis covers the generation of the light at 583 nm and 626 nm,corresponding to the transitions in Er and Dy used for the narrow line MOT, respectively.For the generation of the light at 583 nm we use a seed laser emitting at 1166 nm. This lightis then coupled into a Raman ber amplier and then frequency doubled within a secondharmonic generation (SHG) crystal. With a seed power of 25 mW, we obtain slightly lessthan 2 W of 583-nm light. For the 626-nm light we use two ber lasers at 1550 nm and1050 nm together with two ber ampliers and mix the light in a sum frequency generation(SFG) crystal. With a maximal output power of 5 W for each ber laser we obtain morethan 2 W of light at 626 nm. The light at 626 nm is stabilized with a stable reference cavity.The two mirrors of the cavity and the spacer between them are made of ultra low expansionglass (ULE), which has a measured minimal thermal expansion coecient at 21.78(5) C.The locking of the laser light is done with a ber-coupled electro-optical modulator thatmodulates sidebands at two dierent frequencies. While the laser is resonant to the atomictransition, one of these sidebands is shifted to the cavity resonance. The second sidebandsare then used to generate the Pound-Drever-Hall error signal for the electronic lockingsetup.

4

ZusammenfassungIn dieser Masterarbeit beschreiben wir den Aufbau und die Charakterisierung eines opti-schen Resonators, welcher zur Stabilisierung von Laserlicht verwendet wird. Zum Einenwird damit das Licht, welches für die magneto-optische Falle von Erbium und Dysprosiumbenötigt wird stabilisiert. Zum Anderen soll der Resonator, zu einem zukünftigen Zeitpunktdes Experiments, für die Stabilisierung von Laserlicht für Rydberg Anregungen verwendetwerden können. Des Weiteren behandelt die Masterarbeit die Erzeugung von Laserlicht beieiner Wellenlänge von 583 nm und 626 nm. Diese Wellenlängen entsprechen den optischenÜbergängen, welche für die magneto-optische Falle für Erbium und Dysprosium verwendetwerden. Für das Laserlicht bei 583 nm verwenden wir ein System aus einem Pump-Laser beieiner Wellenlänge von 1166 nm, zusammen mit einem Raman Faserverstärker und einemKristall zur Frequenzverdopplung. Mit einer Pumpleistung von ungefähr 25 mW erhal-ten wir etwas weniger als 2 W Licht bei 583 nm. Für die Erzeugung von Licht bei 626 nmwerden zwei Faserlaser bei den Wellenlängen 1550 nm und 1050 nm mit Faserverstärkernverstärkt und in einem nichtlinearen Kristall für Summenfrequenz addiert. Hier erhaltenwir bei maximaler Leistung der Faserlaser (5 W) eine Ausgangsleistung von etwas mehrals 2 W. Die Spiegel des Resonators und der Abstandshalter dazwischen sind aus einemspeziellen Glas (ultra low expansion glass), dessen thermischer Ausdehnungskoezient beigemessenen 21.78(5) C minimal ist. Zur Stabilisierung des Laserlichts werden mit einemfasergekoppeltem elektro-optischen Modulator Seitenbänder bei zwei unterschiedlichenFrequenzen auf das Licht aufmoduliert. Während der Laser zum atomaren Übergang reso-nant ist, wird ein Seitenband zur Resonatorresonanz geschoben und das zweite Seitenbandzur Erzeugung des Pound-Drever-Hall Fehlersignals verwendet.

6

Contents

1. Introduction 9

2. Overview on the Main Properties of Erbium and Dysprosium 132.1. Erbium and Dysprosium at Glance . . . . . . . . . . . . . . . . . . . . . . . 132.2. Atomic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3. Laser-Cooling Transitions in Erbium and Dysprosium . . . . . . . . . . . . 152.4. Magnetic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3. Laser Systems for Er-Dy Double MOT 213.1. Dual Species Magneto Optical Trap . . . . . . . . . . . . . . . . . . . . . . 213.2. Laser System for 583-nm Light for Erbium . . . . . . . . . . . . . . . . . . 233.3. Laser System for 626-nm Light for Dysprosium . . . . . . . . . . . . . . . . 29

4. Design of the Vacuum Chamber and the ULE Cavity 334.1. Frequency-Stability Requirements of the Light . . . . . . . . . . . . . . . . 334.2. Basic Concepts of Cavity Physics . . . . . . . . . . . . . . . . . . . . . . . 344.3. Cavity Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.4. Mechanical and Electrical Design of the Vacuum Chamber . . . . . . . . . 44

5. Experimental Part 515.1. Optical Setup for the Coupling of the Lasers to the Cavity . . . . . . . . . . 515.2. Modulation-Transfer Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 535.3. Zero Crossing of the ULE Cavity . . . . . . . . . . . . . . . . . . . . . . . . 565.4. Locking of the Laser Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 585.5. Performance of the Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6. Conclusion and Outlook 676.1. Er–Dy Quantum Mixture: Motivation and Vision . . . . . . . . . . . . . . 686.2. Rydberg Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

A. Appendix 71A.1. Pound-Drever-Hall Error Signal . . . . . . . . . . . . . . . . . . . . . . . . 71A.2. Modulation-Transfer Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 74A.3. Electronic Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75A.4. Assembly of the Vacuum Chamber . . . . . . . . . . . . . . . . . . . . . . . 79

Bibliography 85

7

Contents

8

1. IntroductionAt temperatures close to the absolute zero the atoms begin to reveal their quantum behavior.In 1911 Heike Kamerlingh Onnes observed that mercury looses suddenly its electric resistiv-ity at a temperature of 4.2 K [43]. This discovery was related to the rst realization of liquid4He only three years before. The superuidity eect of liquid 4He could be explained in1938 by Kapitza, Allen, and Misener [3, 44]. In the following, in 1933 Walther Meißner andRobert Ochsenfeld discovered that superconducting materials repel an applied magneticeld and are therefore perfectly diamagnetic [66]. It took until 1957, when John Bardeen,Leon Neil Cooper, and John Robert Schrieer found a theoretical microscopic approachto explain the macroscopic eect of superconductivity [9]. However, the understandingof superconductivity changed in 1986, when Johannes Georg Bednorz and Karl AlexanderMueller observed that the critical temperature, where the superconductivity appear, forcopper oxide is much higher than for the materials known up to then. While for othermaterials the critical temperature is below 23 K for copper oxide the critical temperature isat 135 K [10]. Both, superuidity and superconductivity, are macroscopic consequences ofthe bosonic and fermionic character or, more precisely, of the dierent statistical behaviorof the particles. The Bose-Einstein statistics describes particles of whole-integer spin, calledbosons. For zero temperature all identical particles occupy the lowest possible energystate in the system. On the other hand the Fermi-Dirac statistic describes the behavior offermions (particles with half-integer spin) and at zero temperature each energy level upto the Fermi-energy is occupied by one single particle. The resulting states formed by theparticles are called Bose-Einstein condensate for bosons and degenerated Fermi gas forfermions, respectively.

The complex nature of the superconductivity calls for the simulation of the systemsin a better-controllable environment. The development of laser cooling techniques tocool particles with laser light to the µK regime and even lower temperatures lead to newopportunities for the investigation of many-body quantum systems and superuidity.

With the observation of the rst Bose-Einstein condensates (BEC) in 1995 [5, 13, 22],the eld of ultracold atomic gases experienced a fast expansion. A special property ofultracold gases is that they allow a high controllability of internal and external degrees offreedom. Interparticle interactions can be controlled via Feshbach resonances [17], andoptical lattices can be used to create spatial structures. This gives opportunity to enter awide range of physical research elds connected to strongly correlated many-body systemsas Quantum Simulators [30] and high-temperature superconductivity [19].

Among others, important aspects reached by the tunability of the interaction strengthare the crossover from a BEC to a Bardeen-Cooper-Schrieer superuid state [39] and thesuperuid to Mott transition with atoms in an optical lattice [37]. Early experiments withultracold atomic gases haven been realized with alkali atoms as rubidium, lithium, cesium,

9

1. Introduction

and sodium. With the condensation of chromium in 2005 [38] for the rst time an atomicspecies was brought to degeneracy that carries a higher magnetic moment than alkaliatoms, i. e. 6 µB instead of only 1 µB. While for alkali atoms the properties are dominatedby the isotropic and short range contact interaction for dipolar atoms the anisotropic andlong range dipole-dipole interaction (DDI) comes into play. This makes dipolar elementsof special interest for the investigation of new quantum phenomena. With chromium forexample the d-wave collapse of the BEC [48] and the spin relaxation in an optical latticehave been investigated [68].

Since then, BECs and degenerated Fermi gases of erbium and dysprosium that carryeven higher magnetic moments of 7 µB and 10 µB, respectively, have been realized. TheBEC of dysprosium was realized for the rst time in the group of B. Lev [55] at Stanford,whereas the rst realization of an erbium BEC was here in Innsbruck [1]. Dysprosiumshares together with terbium the largest magnetic moment of all elements. While forchromium the contact interaction has to be reduced with a Feshbach resonance in order tomake the DDI dominant, for erbium and dysprosium the DDI is dominant even withoutreducing the contact interaction.

Beside the large magnetic moment, erbium and dysprosium share additional specialproperties, which are the multi-valence electron character and the rich atomic electronicspectra containing many broad and narrow optical transitions [11, 54, 64] that can be usedfor laser cooling to very low temperatures. For both elements the 4 f shell is not completelylled, which results in the high total angular momentum in the ground state. This highangular momentum, the large number of broad and narrow optical transitions, and themulti-valence electron characteristic give new possibilities for the growing eld of Rydbergatoms. These are atoms where one electron is excited to a high principal quantum numbern having a large spacial expansion, e. g. a rubidium atom excited to a state with n = 43 hasan orbital radius that is larger by a factor of 40 compared to the ground state [57]. Thecombination of ultracold gases with Rydberg excitation has already lead to many excitingaspects such as the Rydberg blockade[77, 78] or the spatially ordered structure in a Rydberggas [73]. Similar to atoms with two valence electrons, the core of the Rydberg atom remainsoptically active allowing imaging [65] and manipulation by light scattering [27].

The goal of our experiment is to combine for the rst time two dierent strongly magneticelements, erbium and dysprosium. The experiment will consist of three parts: the mainchamber where the mixture experiments will be done and two additional chambers that willbe attached at a later point. One of these additional chambers will be used to do Rydbergexcitation experiments and for the other one a quantum gas microscope is planed.

About this thesis:The purpose of this master thesis was to work on the generation and stabilization of thecooling light for the narrow-line MOTs of erbium and dysprosium and on the stabilizationof future lasers for Rydberg excitation. For the cooling light for erbium, a seed laser incombination with a Raman ber amplier and a second harmonic generation crystal isused. For dysprosium we use a sum frequency generation crystal to add up the light oftwo ber laser systems consisting of a ber amplier that is seeded by a ber laser. For thestabilization we use a cavity with a spacer out of ultra low expansion glass (ULE) to lock

10

the lasers to the cavity resonance using a Pound-Drever-Hall locking scheme [25]. Theassembling of the cavity and locking of the lasers was the main part of the presented thesis.

The thesis is structured as follows: In Chapter 2 the main properties of erbium anddysprosium such as atomic structure, energy spectra, and magnetic properties are discussed.The topic of Chapter 3 is the generation of the laser light for the MOTs of erbium anddysprosium. This contains a comparison of the parameters used in the already runningexperiments as well as a description of the laser systems that we will use in our experiment.In Chapter 4 a theoretical overview about cavities is given. Further, the requirements forour cavity such as thermal stability are discussed and the nal design of the cavity is shown.The characterization and discussion of the cavity parameters is done in Chapter 5.

11

1. Introduction

12

2. Overview on the Main Properties ofErbium and Dysprosium

In the last years the interest on strongly magnetic lanthanide atomic species increasedsignicantly. These species give access to the study of dipolar phenomena. Recently, erbium(Er) and dysprosium (Dy) have been brought to quantum degeneracy with the creation ofBose-Einstein condensates and Fermi gases [1, 55]. In this chapter some general propertiesof Er and Dy such as level schemes, cooling transitions and magnetic moment are discussed.

2.1. Erbium and Dysprosium at GlanceBoth, Er and Dy are rare-earth elements being part of the Lanthanide series of the periodictable. Er has an atomic number of 68 leading to a mass of 167.27 amu and Dy has an atomicnumber of 66 leading to a mass of 162.5 amu. The melting point of Er is 1529 C, slightlyhigher than for Dy (1412 C). This similarity is in principle an important added valuewhen combining the two species for a quantum mixture experiment. Both, Er and Dyhave dierent fermionic and bosonic isotopes. Er has ve stable bosonic isotopes (162Er,164Er, 166Er, 168Er, and 170Er) and only one fermionic isotopes (167Er). On the other hand Dyhas also ve stable bosonic isotopes (156Dy, 158Dy, 160Dy, 162Dy, and 164Dy) but two stablefermionic isotopes (161Dy, and 163Dy) compared to Er. In Table 2.1 a list of all isotopes,which have a natural abundance above 1 % are listed for Er and Dy.

Table 2.1.:Relative abundances of the stable isotopes of Er and Dy which have a minimum abundanceof 1 %. 167Er, 161Dy and 163Dy are fermionic whereas the other isotopes are bosonic.

Isotope 164Er 166Er 167Er 168Er 170Er

abundance (%) 1.61 33.6 23.0 26.8 15.0boson boson fermion boson boson

Isotope 160Dy 161Dy 162Dy 163Dy 164Dy

abundance (%) 2.3 18.9 25.5 24.9 28.3boson fermion boson fermion boson

A special property of Er and Dy is their large magnetic moment of 7 µB and 10 µB whereµB denotes the Bohr magneton, respectively. Together with terbium, Dy has the largest

13

2. Overview on the Main Properties of Erbium and Dysprosium

magnetic moment of all elements. Table 2.2 gives an overview about some properties of Erand Dy.

Table 2.2.: Overview over some properties of Er and Dy. Z is the atomic number.Note: 1 amu ≡ 1.600 539 040(20) × 10−27 kg [67].

Z mass (amu) melting point (°C) electron cong. mag. momentEr 68 167.27 1529 [Xe] 4 f 126s2 7 µB

Dy 66 162.5 1412 [Xe] 4 f 106s2 10 µB

2.2. Atomic StructureFollowing Mandelung’s rule, the electrons for Er and Dy ll up as

[Xe] 4 f 126s2 (Er) (2.1a)

and[Xe] 4 f 106s2 (Dy). (2.1b)

The notation in square brackets represent the electron conguration of Xenon,

(1s22s22p63s23p63d104s24p64d105s25p6).

While the 4 f -orbital is only partially lled, the 6s-orbital is completely lled up. Thiselectron conguration is referred to as submerged-shell structure. The electrons in the4 f -orbital have an angular momentum quantum number of l = 3. Considering the angularmomentum projection quantum numbers ml = −l, . . . , l and the spin projection numbersms = ±1/2 result in 7 states (ml = −3,−2,−1, 0, 1, 2, 3), which can be doubly occupied.Therefore, in total up to 14 electrons can sit in the 4 f -orbital. The very similar electronconguration of the two atomic species to be stressed whereas Dy has 2 electrons less inthe 4 f -shell compared to Er.

Depending on the spin-orbit coupling energy relative to the Coulomb interaction, dierentcoupling schemes are necessary to nd the respective quantum numbers. If the energyfor the spin orbit coupling Es−o is lower than the residual electrostatic interaction Ere thespin-orbit coupling scheme (LS-coupling1) can be applied. Otherwise, jj-coupling becomerelevant. For Er and Dy only for the ground state the LS-coupling scheme is sucient,whereas for the excited states the jj-coupling has to be applied [80]. For the LS-couplingall spins and angular momenta add up to a total spin ~S =

∑i ~si and a total orbital angular

momentum ~L =∑

i~li. The resulting total spin and total orbital angular momentum couple

then to the total angular momentum, ~J = ~L + ~S . Finally, the quantum state is denoted

1Russell-Saunders coupling scheme.

14

2.3. Laser-Cooling Transitions in Erbium and Dysprosium

by 2s+1LJ . For Er and Dy all electron shells, except the 4 f -shell are completely lled, whichresults in a high total angular momentum in the ground state of J = 6 for Er and J = 8 forDy, respectively.

Following the LS-coupling for the ground state results in 3H6 for Er and 5I8 for Dy. Onthe other hand, for the jj-coupling rst the 4f electrons and the outer electrons coupleindependently via the LS-coupling scheme. This leads to two dierent quantum numbers J1

and J2 which by adding up form the nal state written as (J1, J2)J . As an example one cantake the 583-nm MOT transition of Er, where an electron from the 6s-orbital is excited to a6p-orbital ([Xe] 4 f 12(3H6)6s6p(3P0

1)). First, the inner electrons couple to 3H6 and the outerelectrons couple to 3P1 by LS-coupling. Then, these two states couple to (6, 1)0

7, where thesubscript 7 denotes the sum ~J = ~j1 + ~j2 and the subscript 0 refers to odd party.

The bosons of Er and Dy have no hyperne structure because their nuclear spin I is zero.In contrary, the fermionic isotope 167Er has I = 7/2, which results in a hyperne structurequantum number F = 19/2, whereby the angular momentum J couples to the nuclear spin I.For Dy the nuclear spin is I = 5/2, which leads to F = 21/2 for the electronic ground state.Table 2.3 gives a summary over all quantum numbers in the ground state of Er and Dy.

Table 2.3.: Quantum numbers for the ground state of Er (3H6) and Dy (5I8).

Erbium Dysprosiumtotal orbital angular momentum L 5 6total spin S 1 2total angular momentum J 6 8nuclear spin I for bosons 0 0nuclear spin I for fermions 7/2 5/2

total angular momentum F for fermions 19/2 21/2

Figure 2.1 shows the atomic spectra for Er and Dy. States, which are labeled in red, referto even parity and states in black have odd parity. This very rich atomic spectra containmany dierent possible cooling transitions.

2.3. Laser-Cooling Transitions in Erbium andDysprosium

The plan for the experiment is to have rst a transversal cooling section (TC) to collimatethe divergent beam coming out of the oven. To decelerate the atoms before trapping them aZeeman slower is used (ZS). For the TC and the ZS the broad blue transitions (401 nm for Erand 421 nm for Dy) are used. Since one atom has to scatter many photons over a short timeperiod transitions with a short lifetime are preferred. Additionally, the wavelength shouldbe small, leading to a higher recoil velocity2. For Er (Dy) the transition at 401 nm (421 nm)has a lifetime of 4.94 ns (5.35 ns) and a recoil velocity of 6 mm/s (5.8 mm/s). Assuming, that

2vrec = ~k/m. Here, ~ is the reduced Plack’s constant, m the atomic mass, and kB the Boltzmann constant.

15

2. Overview on the Main Properties of Erbium and Dysprosium

Ferlaino Part B2 RARE

2 3 4 5 6 7 8 9 10 11 120

5

10

15

20

25

4f126s2 3H6

4f12(3H6) 6s6p(3P1)

4f12(3H6) 6s6p(1P1)

03 4 5 6 7 8 9 10 11 12 13

Wav

enum

ber (

1000

cm

-1)

5

10

15

20

25

30

2

4f10(5I8)6s6p(3Po1)(8,1)o

9

4f10(5I8)6s6p(1Po1)(8,1)o

9

4f106s2 5I8 ground state

2000

1000

667

500

400

Wavelength (nm

)333

626 nm (MOT)

421 nm (slowing transition)

401 nm (slowing transition)

421 nm (slowing transition)

583 nm (MOT)

J value

Melting

point (°C)

Slowing transition

[nm] [MHz]

MOT transition

[nm] [kHz]

Isotope

B F

μ (μ_B)

Er 1529 401 30 583 190 5 1 7 Dy 1421 421 32 626 136 5 2 10

J value

ERBIUM DYSPROSIUM

Figure 3: Er and Dy at aglance. (a) Table of the rel-evant propertiesof Er and Dy.Energy level structure of Er (b)and Dy (c) where J is the totalelectronic angular momentumquantum number. States withodd (even) parity are indicatedby black (red) horizontal lines.The ground state of Er (Dy) hasa orbital momentum quantumnumber L = 5 (L = 7). Noneof the bosonic isotopes has anhyperfine structure, whereas thefermionic ones have one. Themain laser cooling transitionsare indicated by arrows. Thepictures are adapted from [50,51].

tube. After the Zeeman slower, the atomic beam will be captured into the magneto-optical trap (MOT)chamber. Both the Er and Dy MOT will operate on their respective narrow inter-combination lines (at 583nm for Er and at 626 nm for Dy); see Fig. 3. The stainless steal MOT chamber will have large viewportsfor optimal optical access and will integrate a multi-channel plate and a appropriate electric field screeningsetup. In this chamber, we will perform evaporative cooling of the mixture in a three-dimensional opticaldipole trap and studies on multi-electron Rydberg atoms. Additionally, the MOT chamber will be connectedto a glass cell, where we plan to implement high-resolution imaging. The glass cell will be screened againstundesired stray magnetic fields by a shielding box made of mu-metal and aluminum.

• Dual species oven: We plan to use a single high-temperature effusion cell to sublimate simultaneously Erand Dy. At an operating temperature of 1100 C, we checked that Er and Dy are not expected to formlow-temperature alloys neither between them nor with tantalum, of which the crucible is made. The slightdifferent of the melting points between the two atomic species will lead to a difference in the vapor pressure.We plan to overcome this problem by using an Er-Dy alloy with the appropriate mixing ratio. The companyACI ALLOYS can produce such a sample.

• Dual species Zeeman slower and MOT: The dual-species Zeeman slower and MOT will operate respec-tively on the strong line and on the inter combination line; see Table X. Given that both species have similarvalues of the mass and similar parameters for the laser cooling transitions, they can be easily slowed downusing the same Zeeman slower. Our preliminary calculations show that a dual-species Zeeman slower of 40-cm length will be sufficient to decelerate the thermal atomic beam from 400 m/s to down to few m/s. Thesmall final velocities are necessary to capture the atoms in our narrow-line MOT. The Zeeman slower tube,including the magnetic field windings, will have a homemade design and it will be built in our mechanicalworkshop.

• Quantum gas microscope: This is a powerful technology, which allows single-particle and spatially re-solved detection in many-body quantum systems. It has been developed by the group of M. Greiner5 [55],and the group of I. Bloch and S. Kuhr [56]. So far it has been achieved only with ultracold alkali atomsin deep pinning optical lattices. Combining this technology with atomic species possessing rich atomiclevel-structure, such as Er and Dy, is a very interesting goal by its own. The variety of possible opticaltransitions will offer many opportunities for the investigation of new high-resolution-imaging schemes. Ofparticular interest is the strong transition in the blue (401 nm for Er and 421 nm for Dy), which is almostan order of magnitude stronger than the alkali one. The quantum gas microscope will be particularly rele-vant in studying the dipolar spinor gas in optical lattices because of the possibility to obtain single-site andsingle-atom resolved images and to reveal the spatial correlations between particles.

• Double-species degeneracy: We plan to operate in double-species configuration and to produce Fermi-5We have an ongoing scientific collaboration with the group of M. Greiner at Harvard regarding high-resolution imaging in lan-

thanide atoms.

8 / 14

Figure 2.1.: Atomic spectra and cooling transitions of Er and Dy. In our experiment we will usethe 583-nm and 626-nm transitions for the Er and Dy MOT whereas the transitions of 401 nm and421 nm will be used for the transversal cooling and the Zeeman slower, respectively. Red linesrepresent states with even parity and black lines states with odd parity. Graphics taken from [8, 81].

Table 2.4.: Electron congurations and quantum numbers of the excited states. First the electronsof the inner and outer shell couple independently via the LS-coupling scheme and then they couplevia the j j-coupling.

electron conguration j j-coupling401 nm [Xe] 4 f 12(3H6)6s6p(1P0

1) (6, 1)07

583 nm [Xe] 4 f 12(3H6)6s6p(3P01) (6, 1)0

7

421 nm [Xe] 4 f 10(5I8)6s6p(1P01) (8, 1)0

9

626 nm [Xe] 4 f 10(5I8)6s6p(3P01) (8, 1)0

9

the atoms coming from the oven have a velocity of approximately 500 m/s, roughly 80 000photons have to be scattered to slow the atoms down to 10 m/s, which is a adequate capturevelocity [33, 61]. The broadness of the transition is a useful characteristic because morephotons can be scattered by the atoms. After the ZS the atoms reach the main chamberwith the MOT. For the MOT we use the 583-nm transition for Er and the 626-nm transitionfor Dy. For both, Er and Dy this means that one electron from the 6s2 state is excited to a6p state and couple with the remaining 6s electron to a triplet 3P1 state. Table 2.4 showsthe electron conguration of the excited states of the laser cooling transitions.

The temperature limit of the Doppler cooling is given by the Doppler temperature3,which is proportional to the linewidth of the transition. The smaller the linewidth, thelower the reachable temperature. The Doppler temperatures for the used transition are

3TD = ~∆ν/2kB. Here, ~ is the reduced Plack’s constant, ∆ν the natural linewidth Γ/2π, and kB the Boltzmannconstant.

16

2.4. Magnetic Properties

TD = 4.6 µK for Er and TD = 3.3 µK for Dy. In principle, also the 401-nm and 421-nmtransitions could be used for the MOT, but due to the relatively high Doppler temperatureof TD = 714 µK for Er and TD = 773 µK for Dy additional cooling steps would have to beimplemented. A low temperature is preferred because of a higher loading rate into a dipoletrap. For laser cooling it is very important to have a close cycling transition. This means,that most of the excited atoms do not get lost in a metastable state, but decay back into theground state. In a metastable state the atoms would became dark to the cooling light. A bigadvantage of the MOT transitions for Er and Dy is that no repumper is needed becausethe decay probability into intermediate states is very low. For Er there are two possibleintermediate states with transition probabilities of 0.017 and 0.0049 [8]. Table 2.5 gives, alist of relevant parameters of the cooling transitions, which will be used in our experiment.

Table 2.5.: Important parameters of the cooling transition of 401 nm, 421 nm, 583 nm, and 626 nm [8,56, 58].

Erbium DysprosiumParameters 401 nm 583 nm 421 nm 626 nm

transition rate Γ (1/s) 1.87 × 108 1.17 × 106 2.02 × 108 0.85 × 106

lifetime τ (ns) 5.35 857 4.94 1170natural linewidth ∆ν (MHz) 29.7 0.19 32.2 0.136saturation intensity IS (mW/cm2) 60.3 0.13 56.3 0.072doppler temperature TD (µK) 714 4.6 773 3.3doppler velocity vD (mm/s) 267 21 198 12.9recoil temperature Tr (nK) 717 339 659 298recoil velocity vr (mm/s) 6.0 4.1 5.8 3.9

The dierent isotopes of Er and Dy have dierent numbers of neutrons and thereforedierent masses. This leads to a dierent distribution of the electrons, resulting in a changeof the charge distribution, causing a shift of the energy levels. Figure 2.2 shows this isotopeshift for the bosonic isotopes of Er and Dy, which have a natural abundance above 1 %.The isotope shift for both elements is on a comparable order.

2.4. Magnetic PropertiesA very special property of these two elements is their large magnetic moment of 7 µB forEr and 10 µB for Dy, where µB denotes the Bohr magneton. This is in contrast to otherelements as alkali atoms, having a 1S 0 state as ground state. The exact value of the magneticmoment µ can be calculated using

µ = mJgJµB, (2.2)

where mJ is the projection of the total angular moment in the direction of a magnetic eldand gJ is the Landé g-factor, also known as the gyromagnetic ratio. The Landé factor canbe calculated by [23]

17

2. Overview on the Main Properties of Erbium and Dysprosium

isotope160 162 164 166 168 170

"8

(GH

z)-1

0

1

2(a)

583 nm626 nm

isotope160 162 164 166 168 170

"8

(GH

z)

-1

0

1

2(b)

401 nm421 nm

Figure 2.2.: The isotope shifts of (a) the 583-nm (Er) and 626-nm (Dy) transitions and (b) the 401-nm(Er) and 421-nm (Dy) transitions [20, 41, 42, 50].

gJ = 1 +J(J + 1) − L(L + 1) + S (S + 1)

2J(J + 1). (2.3)

However, corrections due to deviations from the LS-coupling and other eects, e. g. rel-ativistic corrections, have to be considered. Inserting the Landé factor gJ = 1.163801(1)[21] for Er and gJ = 1.2415867(10) [29] for Dy and assuming that the atoms in the groundstate are sitting in the mJ = 6 or mJ = 8 level leads to a magnetic moment of 6.98 µB and9.93 µB, respectively. By applying an external magnetic eld, the energy level with totalangular momentum quantum number J split up in 2J +1 sub states with magnetic quantumnumber mJ. In Figure 2.3 the Zeeman splitting for ground state of the bosonic isotopes of Erand Dy is plotted for a magnetic eld B up to 5 G. The energy shift can be calculated by

∆E(B) = mJgJµBB. (2.4)

For the fermionic isotopes the hyperne structure has to be taken into account and theunderlying values are the mF state and the Landé g-factor gF . For the lowest hyperne state(F = 19/2 and mF = −19/2 for Er and F = 21/2 and mF = −21/2 for Dy) the magnetic momenthas the same value as for the bosonic ground state.

This large magnetic moment leads to completely dierent scattering properties since thedipole–dipole interaction is fundamentally dierent from the contact interaction. For two

18

2.4. Magnetic Properties

magnetic -eld (G)0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

ener

gy/2:

(MH

z)

-80

-60

-40

-20

0

20

40

60

80 mJ

+8

+7

+6

+5

+4

+3

+2

+1

0

-1

-2

-3

-4

-5

-6

-7

-8

mJ

+6

+5

+4

+3

+2

+1

0

-1

-2

-3

-4

-5

-6

Figure 2.3.: Linear Zeeman energy splitting for the the ground state of the bosonic isotopes of Er(orange) and Dy (blue) with J = 6 and J = 8, respectively. On the right hand site of the graph theprojection quantum number mJ is given.

particles with dipole moments along the unit vectors e1 and e2 the dipole–dipole interactionreads as [49]

Udd(r) =µ0µ

2

4π(e1 · e2)r2 − 3(e1 · r)(e2 · r)

r5 , (2.5)

where r is the relative position of the atoms to each other. For a polarized sample thissimplies to

Udd(r) =µ0µ

2

4π1 − 3 cos2(θ)

r3 , (2.6)

where θ is the angle between the direction of the polarization and the position vector r. Thetwo main properties of Udd are the long-range character, identiable from the factor 1/r3 andthe anisotropy deriving from the symmetry of the Legendre polynomial of second order,P2(cos θ) = 1

2 (1 − 3 cos θ). Since θ can vary between 0 and π/2, the dipole-dipole interactioncan be either repulsive for atoms sitting side by side or attractive for atoms sitting in a

19

2. Overview on the Main Properties of Erbium and Dysprosium

relative angle between two dipoles 3 (:)0 10 20 30 40 50 60 70 80 90

Udd

(4:r3

707

2)

-2

-1.5

-1

-0.5

0

0.5

1

Figure 2.4.: Strength of the dipole-dipole interaction Udd in dependence of the relative anglebetween the dipole orientation at a xed distance r. The green part stands for attractive interactionand the red line for repulsive interaction, respectively. The blue dot marks the so called ’magicangle’, where the dipole-dipole interaction vanishes.

’head to tail’ conguration. Figure 2.4 shows the strength on the dipole–dipole interactionin dependence of the relative angle between the polarization and the relative position of thedipoles at a xed distance r. For the special angle θ = 54.7 °C the dipole–dipole interactionvanishes.

20

3. Laser Systems for Er-Dy DoubleMOT

In this Chapter, we will describe the two laser systems used for the narrow line magneto-optical traps (MOTs) of Er and Dy and the expected requirements for MOT production.

3.1. Dual Species Magneto Optical TrapAmong the dierent possible laser cooling transitions, Er and Dy feature an intercombination-type line, which is broad enough to provide ecient laser cooling but narrow enough tohave conveniently low Doppler temperatures. This is the case of the transition at 583 nm forEr and at 626 nm for Dy. For both species, single-species MOTs are currently successfullyoperating on such transitions [24, 33, 61]. The relatively narrow linewidths of aroundΓ ≈ 2π · 150 kHz arise as a result of a transition from the ground state being a singlet stateto an excited state being a triplet state (intercombination line).

The experimental sequence in the experiments, where the narrow line MOTs are realized,is as follows: First, the atoms are loaded from the Zeeman slower (SL) into a MOT withlargely detuned laser light compared to their natural linewidth, i. e. approximately −40 Γ583

for Er and −35 Γ626 for Dy, respectively. Due to the large detuning the atoms do not feela homogeneous light force over the volume of the MOT but are resonant with the lighton an ellipsoid of constant magnetic eld [32]. Due to the gravitational force, the atomswill accumulate in the lower region of this ellipsoid and form a characteristic bowl-shapedMOT. This automatically prevents the Zeeman slower light to reduce the atom number inthe MOT by exerting a light force on the atoms. Since for the double species MOT bothatomic species feel the same magnetic eld gradient, the position relative to each other canbe inuenced by the detuning. In a second step the MOT is compressed (cMOT) to reachlower temperatures and a higher density of the atomic cloud. During the compression themagnetic eld gradient is changed as well as the detuning and the light intensity. Figure 3.1shows a schematic representation of the passage from a MOT to a cMOT for the simplecase with a J = 0→ J′ = 1 transition. Table 3.1 gives some typical parameters used in [61]and [33] for the realization of the narrow line MOTs.

In summary this narrow line transitions provide a exceptionally good starting point forthe subsequent direct loading of a dipole trap due to the already low nal temperatures andlarge atom densities reached after the cMOT step. Both species, Er and Dy reach similarnal temperature, atom number, and atom density for comparable detunings and magneticeld gradients. This leads to the promising chance of realizing a simultaneous doublespecies MOT using adequate trade-o parameters especially for the magnetic eld gradient.

21

3. Laser Systems for Er-Dy Double MOT

mJ = +1

mJ = -1

mJ = +1

mJ = -1

small δ

large δ

z

E

Figure 3.1.: Representation of a transition from a MOT to a cMOT. For the cMOT the magnetic eldgradient, the beam intensity and the detuning are reduced compared to the starting MOT.

Table 3.1.: Summary of the parameters used for the narrow line MOTs in [61] and [33].

Erbium Dysprosium

transition 583 nm 626 nmnatural linewidth Γ 0.19 MHz 0.136 MHzmagnetic eld gradient MOT 2.6 G/cm 3 G/cmdetuning MOT −40 Γ583 −35 Γ626

magnetic eld gradient cMOT 0.9 G/cm 1.5 G/cmdetuning cMOT −9.5 Γ583 −2.75 Γ626

nal temperature 10 µK 6 µKnal atom number 2 × 108 1.5 × 108

nal atom density 1.5 × 1011 /cm3 8.6 × 1010 /cm3

An issue that could have a negative inuence on the double-species MOT are possible crosstalks. This means that the light at 583 nm for the Er MOT could have an eect on the Dyatoms and the light at 626 nm could have an eect on the Er atoms. Table 3.2 and 3.3 givethe closest known transitions with respect to the MOT transition of each species.

Table 3.2.: Closest Er transitions near Dy MOT line.

J → J′ λ (nm) ∆ν = Γ/2π (kHz)6→ 6 622.102 133.76→ 6 620.2 55

Table 3.3.: Closest Dy transitions near Er MOT line.

J → J′ λ (nm) ∆ν = Γ/2π (kHz)8→ 7 565.36 718→ 8 597.6 678→ 7 599.02 90

22

3.2. Laser System for 583-nm Light for Erbium

time (1/50ms)

volta

ge (1

/40m

V)

(a) Acoustic noise on the laser of 1166 nm.

time (1/100ms)

volta

ge (1

/20m

V)

(b) The acoustic noise is removed with the ex-ure laser design.

Figure 3.2.: Figure (a) shows the acoustic noise on the light of 1166 nm. The noise was removed (b)by changing the laser design to the exure design. Blue is the voltage applied for the piezo scan,orange is the cavity transmission, and in (a) purple is the cavity reection. The scan ramp is notsymmetric to the cavity transmission.

The nal performance of the double-MOT will also crucially depend on interspeciesinelastic collisions, which have to be investigated experimentally [36].

3.2. Laser System for 583-nm Light for ErbiumTo create the 583-nm light for the narrow line MOT of Er we use a commercial Ramanamplier from MPB Communications, which is seeded from a homemade grating-stabilizedexternal cavity diode laser (ECDL) at 1166 nm.

External cavity diode laser at 1166 nmAs already mentioned above we plan to use a home built ECDL as a seed laser. There areonly a few laser diodes available covering the needed wavelength of 1166nm. We chosea high power gain chip1 based on an InAs Quantum Dot with a central wavelength at1178 nm and a tuning range of 50 nm.

Initially, we used a laser design, which was frequently used in Innsbruck, e. g. in [76].However, we observed a high sensitivity to acoustic noise. Figure 3.2(a) shows the cavitytransmission of the 583 nm while scanning the laser over the cavity resonance. Althoughhigher transversal modes are suppressed by the coupling, many other modes appearedwhen clapping or knocking on the laser table. Changing to a exure-based laser design [47]solved this issue. Figure 3.2(b) shows the cavity transmission of the 583 nm laser with theexure-based laser as a seed laser. The laser design works in Littrow-conguration, where

1GC–1178–TO–200 from Innolume, 200 mW @ 1178 nm.

23

3. Laser Systems for Er-Dy Double MOT

the 1th diraction order of the grating is reected into the gain chip and the 0th order is usedas an output. In comparison to the rst Innsbruck design the diraction grating and theoutput coupling mirror of the new design are glued on a compact exure instead of a mirrormount. Figure 3.3 shows the design of the exure laser. The gain chip is mounted in a

diode holder

screws to turn the flexuregrating

out coupling mirror

flexurePeltier element

screws for vertical adjustment

piezo

Figure 3.3.: Drawing of the exure type laser design. The gain chip is xed in the diode holder,which is temperature stabilized by the Peltier element. Two screws are placed in the diode holderfor the vertical adjustment. For rough wavelength adjustment the exure (blue) is turned with thebrown screws on the side. The piezo element is for the ne adjustment of the wavelength.

standard collimation package from Thorlabs2 which is used for collimation of the light. Thecollimation package tube sits inside an aluminum holder which gets temperature stabilizedvia a Peltier element3, see Fig. 3.3. The heat from the Peltier elements is removed via a heatsink, which is glued on the Peltier element and connected to the housing. The temperaturestability is further improved by stabilizing the laser housing itself using additional Peltierelements sitting between the laser housing and the baseplate.

The wavelength of the laser is adjusted by turning the exure, whereby the grating4 ischosen in such a way, that the angle between the 0th order and the 1th order is approximately45 degree. This ensures that the light leaves the laser without hitting the housing. The

2e. g. Thorlabs LT230P-B.3e. g. Peltier element with 5.1 W, 2.2 A, and 3.8 V.4For 1166 nm the grating GH13-12V from Thorlabs is chosen.

24

3.2. Laser System for 583-nm Light for Erbium

rough wavelength adjustment is done by turning the exure via two adjustment screws thatpress on the exure. The ne adjustment as well as the scanning of the wavelength is doneusing a piezo element placed in the exure and xed with a screw that presses on the piezo.The special structure in the exure allows to turn the grating without turning the wholeexure when applying a voltage to the piezo. The laser housing provides additional spacefor a lter- or prestabilization cavity and the possibility to evacuate the whole housing foradditional temperature- and acoustic shielding. However, the linewidth and acoustic noiseshielding are already sucient without those options.

Figure 3.4 shows the characterization of the achievable output power of the 1166 nmlaser for three dierent temperatures. For a better clearness the data points are shownwithout error bars. The error can be assumed with approximately 5 %. The threshold of

current (mA)50 100 150 200 250 300 350 400 450

outp

ut p

ower

(m

W)

0

20

40

60

80

100

120

140

T = 21/CT = 22/CT = 23/C

Figure 3.4.: Output power of the 1166 nm laser as a function of the current for three dierenttemperatures. Error bars are not shown for clarity.

the gain chip was between 109 mA for 21 C and 115 mA for 23 C. Although the gain chipis specied up to 800 mA the output power shows a saturation after 400 mA. Since only20 mW of 1166-nm light are needed for seeding the Raman ber amplier we restrictedthe the maximal current used to below this value to avoid this problem. Unfortunately,the Quantum Dot chip behaves in frequency domain dierently than what expected (seediscussion below). Therefore, we nally decided to buy a commercial system as a seed laserat 1166 nm.

Raman fiber amplifierThe commercial system consists of three elements:

• Ytterbium Fiber Laser

25

3. Laser Systems for Er-Dy Double MOT

• Raman Fiber Amplier

• Second Harmonic Generator

The working principle of the amplier is based on induced Stokes-Raman-scattering, whereboth a pump photon and a photon of the desired frequency enter the optical material.For the Raman ber amplier the optical material is based on a single-mode polarization-maintaining ber. The Stoke-Raman-scattering describes a photon at a certain frequency ν1

interacting with the material and leaving either with lower frequency νs = ν1 − νR (Stoke-Raman-scattering) or with higher frequency νA = ν1 + νR (anti-Stoke-Raman-scattering). νRis the frequency of an oscillation- or ration mode of a molecule or a solid. The enteringphoton either transfer energy to the medium or gets energy from the medium. The inducedStoke-Raman-scattering occurs when a photon enters the medium together with a pumpphoton of higher frequency (Stoke-Raman-scattering). The photon induces the emission ofa second photon, which has the same frequency as the rst one. In our case, this leads toan amplication of the seed light by about a factor of ≈ 300.

After the amplication stage, the light gets frequency doubled in a Second HarmonicGenerator (SHG), which in this case uses a periodically-poled lithium niobate crystal(PPLN) in single-pass conguration. The special property of a non linear crystal is that thepolarization of the medium does not depend linearly on the electric eld. The polarizationof the material can be written as Taylor expansion. For a non linear crystal, where higherorders except the second can be neglected, the non-linear polarization of the medium independence of the electric eld E = ReE(ω) exp(iωt) of the incident light can be writtenas

Pnl = χ(2)E2. (3.1)

χ(2) is the second order electric susceptibility, describing the strength of the non linearresponse. In general, χ(2) is a tensor of 3th rank. However, for simplicity we take Pnl andE to be scalar quantities. Inserting the electric eld E = ReE(ω) exp(iωt) gives for thepolarization

Pnl(t) = Pnl(0) + RePnl(2ω) exp(i2ωt),

which consists of the linear term Pnl(0) and the frequency doubled term RePnl(2ω)ei2ωt.PPLN is a material engineered being quasi-phase-matched. This means, that the orienta-

tion of the electric susceptibility of the crystal is inverted periodically. This avoids that thegenerated photons run out of phase to each other and interfere destructively. The neededperiod of the alternating crystal is set by the desired wavelength to be generated. Thedoubling eciency, according to the company, is about 30 %.

Figure 3.5 shows the optical setup of the home made 1166 nm seed laser together withthe commercial Raman ber amplier and the SHG crystal. An optical isolator5 is placed toprevent that light is back reected into the laser and disturbs the behavior of the gain chip.The 1166-nm light is fed into the Raman ber amplier and transported to the SHG crystalvia optical bers. Afterwards, the 583-nm light is spitted into three paths, which go to the

5Isowave Manufacturing: I-11B-4.

26

3.2. Laser System for 583-nm Light for Erbium

experiment, the cavity, and the wavemeter. The intensity of the 583-nm light, which goesto the experiment is stabilized with an acousto-optical modulator6 (AOM).

EOM

lensf = 50mm

PD

PBS PBSHCL

λ/2

λ/2

λ/2

fiber outputcoupler

mixerLO 5 MHz

EO

M

EO

M

PDPD

lensf = 50mm

lensf = 50mm

BS

BS

fiber outputcoupler

fiber outputcoupler

lensf = 500mm

M1 M1

M2

M2

dichroic mirror

PD

PID

Laser

BS

PBSexperiment

λ/2

cavity

cavity

fmod + fshiftfmod

583 nm

626 nm

1050

nm

1550

nm

PPN

L

f = 50mmf = 100mm

f = 200mm

f = 50mm

f = 100mmλ/2λ/4

λ/4λ/2

f = 200mm

f = 150mm

λ/2

λ/2

PBS

PBS

λ/2

experiment

wavemeter

cavity

spectroscopy signal

EO

M

ECDL

λ/2

λ/2

PBS

AOM

SHG

λ/2 λ/2

λ/2PBS

PBS

AOM

fiber laser+

fiber amplifier

1166 nm

583 nm

wavemeter

cavity wavemeterwavemeter

experiment

f = 175mm

f = -50 mm

f = 200 mm

f = 100 mm

f = -50 mm

f = 50 mm

f = -30 mm

optical isolator

Figure 3.5.: Drawing of the optical setup for the generation of the 583-nm light. The setup is dividedinto the seed laser at 1166 nm and the SHG crystal with the 583 nm light.

Figure 3.6(a) shows the output power of the 583-nm light for dierent pump laser currents.The current is representative for the pump power. The output power also depends on thetemperature of the crystal as the phase matching condition for the SHG crystal has tobe fullled. As the temperature inside the crystal changes with the total power, the settemperature of the SHG crystal has to be adjusted for each current to maximize the outputpower. Figure 3.6(b) shows the output power as a function of the crystal temperature whenthe pump power was kept constant (4.5A). To extract the temperature at which the outputpower is maximal we t P(T ) = A + B · sinc2(C(T −D)) to the data points. The t describesthe central part well and we obtain a maximum power at a temperature of 64.97(2) C. Theidea for the locking of the laser is to generate the error signal (see Sec 5.4) with the 583-nmlight and feed the error signal to the 1166-nm ECDL.

6Gooch & Housego, 3080-215.

27

3. Laser Systems for Er-Dy Double MOT

current (mA)0 1000 2000 3000 4000 5000 6000 7000 8000

pow

er 5

83 (

W)

0

0.5

1

1.5

2

2.5

(a) Power of the 583-nm light as a function of the current of the pump laser.

60 61 62 63 64 65 66 67

pow

er 5

83 (

mW

)

0

50

100

150

data pointsA + B " sinc2(C(T !D))

(b) Power of the 583-nm light as a function of the temperature of the SHG crystal.

Figure 3.6.: Output power of the system for the generation of 583 nm light. (a) shows the outputpower of 583-nm light vs. the current of the Ytterbium ber laser. In (b) the output power of 583-nmlight vs. the temperature of the SHG crystal is plotted. The amplier current was at 4.5 A. For bothmeasurements the seed power at 1166 nm was at 25 mW.

28

3.3. Laser System for 626-nm Light for Dysprosium

Issues with the Seed Laser at 1166 nmAfter coupling the 583-nm light into our home built cavity (see Chap. 4) we observed thatthe laser was not always running single mode and sensitive to the current. Already a veryslight change of the current lead to a mode-hop or a change from single-mode to multi-mode.Furthermore, it happened that after running single mode for some time the laser jumpsinto multi mode. The minimum time between we observed being single mode and multimode was only a few minutes, whereas the maximum time observed was 1 to 2 hours. Thecurrent plateau, where the laser was single mode, was determined to be 0.6 mA, whichmakes single-mode operation at the desired wavelength very challenging. Additionally, itwas observed that the long term stability of the wavelength was not satisfying. To test this,the laser was kept switched on for one night, whereby the next morning the laser was multimode and it was not possible to get the laser back to the desired wavelength by adjusting thecurrent and the temperature. A priori it was not clear if the laser diode itself, the electronicsfor the diode current or the temperature stabilization are responsible for this behavior. Aftercareful testing of dierent electronics and checking the temperature stability, we concludedthat the laser diode itself was responsible for the frequency instabilities. A possible solutionfor this issues is to use a dierent type of gain chip. For an ECDL, two types of gain chip areavailable: type A and type B. While type A has a straight stripe normal to the facets typeB has a curved stripe. The tilted stripe has the advantage of reducing the self lasing andminimizing gain ripples. Another improvement would be the reduction of the cavity lengthof the ECDL since this would increase the current plateau where the laser runs single-mode.

From the experiences of previous experiments [33] an optimal intensity for the narrowline MOT of Er is 12 × Isat, which leads to a total power requirement of

P583 = 12 · Isat · A · 3 ≈ 60 mW, (3.2)

where the factor 3 refers to the three MOT beams (we use them in retro reecting con-guration), A = π · 4 cm2 is the beam area with a beam waist of 2 cm and the saturationintensity Isat = 0.13 mW/cm2 (see Table 2.5). This value for the power is just a referencevalue estimated with the experience of the running experiment. The exact necessary powerhas to be determined empirically.

3.3. Laser System for 626-nm Light for DysprosiumThe laser system we use for the creation of 626-nm has been implemented successfullyin dierent experiments, e. g. in the groups of D. J. Wineland at NIST [79] and T. Pfau inStuttgart [61]. The light is generated via sum frequency generation (SFG) of 1550-nm7 and1050-nm8 light in a periodically poled lithium niobate (PPLN) crystal9. Fig. 3.7 shows theoptical setup for the generation of the light 626 nm. An AOM10 stabilizes the intensity,which goes to the experiment. The two laser systems consist of a seed laser and a ber

7NKT Photonics: Koheras Boostik HPA Y10 System.8NKT Photonics: Koheras Boostik HPA E15 System.9Covesion: MSFG626-0.5-40.

10Gooch & Housego, 3080-215.

29

3. Laser Systems for Er-Dy Double MOT

EOM

lensf = 50mm

PD

PBS PBSHCL

λ/2

λ/2

λ/2

fiber outputcoupler

mixerLO 5 MHz

EO

M

EO

M

PDPD

lensf = 50mm

lensf = 50mm

BS

BS

fiber outputcoupler

fiber outputcoupler

lensf = 500mm

M1 M1

M2

M2

dichroic mirror

PD

PID

Laser

BS

PBSexperiment

λ/2

cavity

cavity

fmod + fshiftfmod

583 nm

626 nm

1050

nm

1550

nm

PPN

L

f = 50mmf = 100mm

f = 200mm

f = 50mm

f = 100mmλ/2λ/4

λ/4λ/2

f = 200mm

f = 150mm

λ/2

λ/2

PBS

PBS

λ/2

experiment

wavemeter

cavity

spectroscopy signal

EO

M

ECDL

λ/2

λ/2

PBS

AOM

SHG

λ/2 λ/2

λ/2PBS

PBS

AOM

fiber laser+

fiber amplifier

1166 nm

583 nm

wavemeter

cavity wavemeterwavemeter

experiment

f = 175mm

f = -50 mm

f = 200 mm

f = 100 mm

f = -50 mm

f = 50 mm

f = -30 mm

optical isolator

Figure 3.7.: Optical setup for the SFG of the 626-nm light. The 626-nm light is splitted into threepaths: one is going to the experiment, one to the ber coupled EOM/cavity and one goes to thewavemeter.

amplier. The ber amplier for the 1550-nm system is a Er doped ber laser, whereasthe ber amplier for 1050-nm is ytterbium doped. The laser systems provide an outputpower of 5 W each and have a free running linewidth of < 1 kHz (1550 nm) and < 20 kHz(1050 nm). Both lasers can be tuned with a piezo element that has a modulation frequencylimit up to 20 kHz. To lock the 626-nm we couple it into the ULE cavity and generatea PDH error signal, which we feed back to the 1050-nm seed laser while the system of1550 nm remains free running (linewidth < 1 kHz). This is possible, because the lasers havea large mode-hop-free detuning range (10 GHz for 1050 nm and 8 GHz for 1550 nm) and asucient long term stability. Since the linewidth of the laser systems are small enough,the locking only needs to account for long term drifts. A more detailed description of thelooking scheme is given in Sec. 5.4.

To generate the 626-nm light as ecient as possible the phase matching condition~k626 = ~k1550 + ~k1050 has to be fullled for the SFG crystal. The phase matching conditionis fullled via the PPLN crystal, as explained already for the SHG crystal in Sec. 3.2. For theSFG crystal three dierent periodicities are available within one single crystal, 11.12 µm,11.17 µm, and 11.22 µm, whereby we use the 11.22 µm. The quasi-phase-matching can beinuenced slightly by varying the temperature of the crystal. In Figure 3.8(a) the powerof the 626-nm light is plotted against the temperature of the PPLN crystal and tted with

30

3.3. Laser System for 626-nm Light for Dysprosium

P(T ) = A + B · sinc2(C(T − D)). Although the data points are asymmetric, the sinc2

describes the central part suciently good to extract the temperature with maximal powerto 170.00(1) C.

Additionally to the phase-matching, the intensity of the pump beams is important. There-fore, the 1550-nm and the 1050-nm light is focused into the crystal fullling approximatelythe condition z0 = l/2, where z0 is the Rayleigh length and l the length of the crystal. This is arule of thumb recommended by Coevision. Our crystal has the dimensions (40×10×0.5) mm.

Similar to the SHF in Sec. 3.2, the SFG is based on the polarization P of the non-linearcrystal. Inserting Einc = E(ω1)eiω1t + E(ω2)eiω2t in Eq. 3.1 leads to

PNL(ω1 + ω2) = χ(2) · E(ω1)E(ω2)ei(ω1+ω2t), (3.3)

oscillating at ω1 + ω2. Therefore, the amplitude of PNL(ω1 + ω2) depends on the product ofthe amplitudes of the two incoming waves. In Figure 3.8(b) the product of the power of thepump lasers is plotted against the power of the 626-nm light. The conversion eciencycan be determined from the slope of a linear t (B · x), to the data points to

ηconv = 9.48(34) %

As for the 583-nm light the optimal intensity for the MOT can be estimated from theexperience of a running experiment. In [24] a power of ≈ 50 · Isat is used for each beam,leading to

P626 = 50 · Isat · A · 3 ≈ 135 mW, (3.4)

where Isat = 72 µW/cm2 is the saturation intensity and A = π · 4 cm2 is the beam areawith a beam waist of 2 cm. The factor 3 arises from the three beams in retro reectingconguration, which we will have for the MOT. This value is just a reference estimatedwith the experience of the running experiment [24]. The nal necessary power has to bedetermined by optimizing the MOT loading phase in the experiment.

31

3. Laser Systems for Er-Dy Double MOT

temperature (/C)165 166 167 168 169 170 171 172 173 174 175

pow

er(W

)

0

0.5

1

1.5

2

2.5

data pointsA + B " sinc2(C(T !D))

(a) Output power vs. Temperature of the SFG crystal at an amplier current of I1550 = 8.3 A(I1050 = 6 A) and a seed laser output power of P1550 = 6.7 mW (P1050 = 6.6 mW).

P1550 " P1050 (W2)0 5 10 15 20 25

P626

(W)

0

0.5

1

1.5

2

2.5

(b) The output power of 626-nm light is plotted versus the product of the power of the 1550 nmand 1050 nm laser (circles). The data points are tted with a linear function B · x.

Figure 3.8.: Output power of the system for the generation of 626 nm light. (a) shows the outputpower of 626-nm light in dependence of the temperature of the PPLN crystal. (b) shows the outputpower of 626-nm light in dependence of the product of the power of the 1550 nm and 1050 nm laser.

32

4. Design of the Vacuum Chamberand the ULE Cavity

The present Chapter describes the core project of my master thesis. Indeed, the projecttargets to build a stable cavity to lock narrow-line laser beams. As discussed in Chap. 3, thecavity will be used for the Er and Dy MOT light. In addition, our experiment also aims atstudying Rydberg physics in Er. To excite ground-state atoms to a Rydberg level, we planto use a two photon excitation, both in the UV. For ecient transfer, the laser light for theexcitation needs to have small linewidth. Our cavity is indeed designed for light for theMOTs as well as for the Rydberg excitation light.

In this chapter, rst, we discuss the frequency stability of light and give a short overview ofthe theoretical aspects regarding the cavity. Afterwards, we discuss the cavity requirementsand describe the setup of the vacuum chamber for the cavity.

4.1. Frequency-Stability Requirements of the LightThe narrow line magneto optical traps (MOTs) of Er and Dy [33, 61] as well as Rydbergexcitation schemes require lasers with a high frequency stability. The stability of a laseris described by the frequency jitter, which can be divided into fast and slow jitter. Thedivision between slow and fast jitter depends on the time scale. The fast jitter is practicallynot resolvable on the respective timescale and corresponds therefore to the linewidth ofthe laser light. For the short term stability of the laser the fast jitter is relevant and thewidth in frequency should be smaller than the linewidth of the transition. For Example, ifthe linewidth of the laser light for the MOT is broader than the transition linewidth of theatom, it is possible, that the necessary detuning is not guaranteed anymore. In [16] it isshown, that trapping of atoms with a natural linewidth of ≈ 6 MHz with laser light with alinewidth of 10 MHz leads to a 40 % lower atom number compared to trapping with laserlight with a linewidth of 1 MHz.

The linewidths of the cooling transitions for Er and Dy are on the order of 100 kHzand 2 kHz–10 kHz (see Table 2.5 for transition parameters), whereby the linewidths for thelasers for Rydberg excitation are on the low kHz level [57]. For laser cooling the time scalefor the fast jitter is in the range of 1 µs to 1 ms, because it is the time scale where the atomsscatter photons.

The slow jitter can in practice be resolved and is therefore referred to as frequency driftand related to the long term stability. In our case, the frequency drift over one day should besuch low that the laser remains resonant to the atomic transition. This means, the laser issatisfying stable if the frequency drift over one day is on the order of a few kHz. If the long

33

4. Design of the Vacuum Chamber and the ULE Cavity

term stability of the lasers is unsatisfying, the lasers would have to be readjusted regularly.One possibility to stabilize the lasers to the desired wavelength is to use a lock to a

stable reference cavity. For example the frequency drift of a similar cavity [46] over 7 hwas estimated to 2.77 Hz/s leading to a satisfying long term stability. Cavities can also beused to improve the short term stability, as it has been shown in [71], where for a dye laserat 583 nm a linewidth of 45 kHz was obtained and in [53] the linewidth of 200 kHz of a741 nm external cavity diode laser was reduced to 1 kHz.

4.2. Basic Concepts of Cavity PhysicsThe aim of this section is to give a brief overview of the theoretical basics of a Fabry-Perotcavity formed by two at mirrors as illustrated in Fig 4.1. A more detailed description canbe found in many textbooks, e. g. [72]. For the derivation of the most important propertiesfor this thesis it is sucient to concentrate on ray optics describing the incoming wave as

Einc = E0eiωt, (4.1)

with ω being the laser frequency and E0 the complex amplitude. The electric eld picks up

Einc

t Einc

t r Einc

t2 r Einc

t2 r2 Einct3 r2 Einc

t,r t,r

Figure 4.1.: Fabry-Perot cavity and electric eld of the incoming beam. t and r label the transmissionand reection coecient, respectively.

an additional phase while traveling between the mirrors which can be expressed as

∆φ = ω(n + 1)L

c, (4.2)

where L is the distance between the mirrors of the Fabry-Perot cavity, c is the speed oflight and n labels the number of reections before leaving the cavity. To obtain the totaltransmitted light eld, the partially transmitted elds have to be summed up as

Etrans = Einc(tei∆φ + r2tei3∆φ + r4tei5∆φ + · · ·

), (4.3)

where r is the reection coecient and t the transmission coecient of the mirrors. Forthe derivation we assume perfect mirrors neglecting any losses. Both coecients are thenconnected via

r2 + t2 = 1. (4.4)

34

4.2. Basic Concepts of Cavity Physics

Using the geometric series and taking the square of the transmitted electric eld leads tothe transmitted intensity

Itrans = IincT 2

T 2 + 4R sin2(∆φ). (4.5)

In the same way, the reected intensity can be written as

Iref = Iinc − Itrans = Iinc4R sin2(∆φ)

T 2 + 4R sin2(∆φ), (4.6)

where T = t2 is the transmission and R = r2 is the reectivity. The transmission signal of acavity is plotted in Fig. 4.2 for two dierent reectivity values.

frequency/FSRn-2 n-1 n n+1 n+2

rela

tive

inte

nsity

0

0.2

0.4

0.6

0.8

1Cavity Transmission

r = 0.9r = 0.99

FSR

"8

Figure 4.2.: Theoretical plot for the transmission of a cavity. The signal in blue is for a reectivityof 0.9, whereas the red signal is for a reectivity of 0.99. The spacing between two cavity-modescan be expressed as νn − νn−1 = c

2L and is equal to the FSR.

Important parameters of a cavity are the free spectral range (FSR) and the full width athalf maximium (FWHM). The FSR

FSR =c

2L(4.7)

describes the distance in frequency between two neighboring longitudinal modes. Thefull width at half maximum (FWHM) corresponds to the linewidth ∆ν of the transmissionpeaks. Assuming low losses in the cavity it can be calculated as

∆ν =FS Rπ

arccos[1 −

(1 − R)2

2R

]. (4.8)

An additional important parameter to characterize a cavity is the nesse F , which simplyis the ratio between the FSR and the FWHM:

F =FSR

FWHM . (4.9)

35

4. Design of the Vacuum Chamber and the ULE Cavity

For the special case of low losses and high reectivity as in our case, the nesse can beexpressed as

F =π√

R1 − R

. (4.10)

The linewidth of the resonance can be interpreted as a decay of optical energy due toresonator losses and is therefore related to the average photon lifetime, τP, in the resonator:

τP =1

2π∆ν, (4.11)

where ∆ν is the linewidth of the cavity. If we consider wave optics instead of beam optics,beside the longitudinal modes, only diering in frequency, also higher transversal modesexists, which dier in their intensity pattern and can be dierent in frequency. In waveoptics the solution of a resonator is described by the Hermite-Gauss functions, written asTMEnm with the intensity distribution

I(x, y, z) = I0

Hn

√2w(z)

exp(−x2

w(z)2

)2 Hm

√2w(z)

exp(−y2

w(z)2

)2

,

where Hk is the kth Hermite polynomial, w(z) is the waist (see below), and n and m arepositive integers. In Figure 4.3 the intensity distribution in the x-y plane for dierent TMEmodes is plotted. The TME00 is called the Gaussian mode and is the only mode which hasthe same resonance conditions as obtained with ray optics, whereas higher TME modeshave dierent resonance conditions depending on the cavity geometry. Those highercavity modes can have a negative eect on the locking if they are to close in frequencyto the TME00 mode and therefore inuence the PDH-error signal (see App. A.1). For thegeometrical case in which one cavity mirror is at whereas the other mirror is spherical,the frequency dierence between two TMEnm modes is given by [28]

δν =c

2L(n + m + 1)

πarccos

√

1 −Lρ

, (4.12)

where L is the length of the cavity and ρ is the radius of the spherical mirror. For ourcavity parameters of L = 0.15 m and ρ = −500 mm this leads to δν ≈ 210 MHz. Since wewill lock the laser to the TME00 mode it will be important, that the adjacent cavity modesare far enough away from the sidebands for the generation of the PDH error signal (seeSec A.1). Otherwise, the latter could be disturbed by the close lying cavity modes. Sinceour FSR equals approximately 1 GHz, cavity modes with m + n ≥ 5 have to be sucientlysuppressed. This is easily possible with a good cavity coupling.

For the description of the coupling to a cavity the laser beam has to be treated as aGaussian beam, which can be characterized by the q-parameter

1q

=1

R(z)−

iλπW(z)2 , (4.13)

36

4.2. Basic Concepts of Cavity Physics

x (nm)-20 0 20

y(n

m)

-20

-10

0

10

20

(a) TME00

x (nm)-20 0 20

y(n

m)

-20

-10

0

10

20

(b) TME01

x (nm)-20 0 20

y(n

m)

-20

-10

0

10

20

(c) TME11

x (nm)-20 0 20

y(n

m)

-20

-10

0

10

20

(d) TME22

x (nm)-20 0 20

y(n

m)

-20

-10

0

10

20

(e) TME04

x (nm)-20 0 20

y(n

m)

-20

-10

0

10

20

(f) TME33

Figure 4.3.: The intensity distribution of dierent transversal cavity modes is plotted.

37

4. Design of the Vacuum Chamber and the ULE Cavity

containing the curvature

R(z) = z[1 +

(z0

z

)2]

(4.14)

and the waist of the beam

w(z) = w0

√1 +

(zz0

)2

, (4.15)

where λ is the wavelength, z is the expansion direction of the beam, w0 is the minimumwaist in the focus of the beam and z0 is the position where the curvature of the beam ismaximal, respectively. To couple a laser beam to a cavity, it is important that the beamtravels exactly along the axis of the mirrors and that the curvature of the wavefront ofthe beam equals the radius of the mirrors. Since we use one planar mirror and one mirrorwith a radius of ρ = −500 mm the beam has to be focused on the planar mirror and bycombining Eq. 4.14 and Eq 4.15 one obtains that the beam waist in the focus W0 has to be

w0 =Lλπ

√ρ − L

L. (4.16)

By inserting our cavity parameters, L = 15 cm and ρ = −500 mm, we obtain for example abeam waist of W0 = 214 µm for λ = 626 nm. In Sec. 5.1 the optical setup for the coupling tothe cavity is explained.

4.3. Cavity Requirements

Reflectivity RequirementsAn important requirement for our cavity is the possibility to frequency stabilize light atseveral wavelengths. On the one hand, we want to stabilize the light for the MOT of Er andDy and, on the other hand, we want to lock the light for Rydberg excitation. Table 4.1 givesa list of all required wavelengths or wavelength ranges. The 583-nm transition for Er andthe 626-nm transition for Dy are used for the MOTs, respectively. The 631-nm transitionand the 741-nm transition are even narrower transitions, that can be used for additionalcooling [56] or state preparation. The exact wavelengths for the Rydberg excitations arenot known yet, as many dierent excitation schemes are possible. For this reason a broadwavelength range was chosen. Furthermore, the largest isotope shift for Er and Dy is about1 GHz (see Fig. 2.2), that suggests to use a cavity with L = 15 cm, which leads to a FSRof 1 GHz. This makes the switching between the isotopes easier because the laser can belocked to the next cavity resonance (see Sec. 5.4).

The multiwavelenght operation requires to employ cavity mirrors with high reectivityover a wide frequency range. The reectivity of a mirror is related to the Bragg diractioncondition. For high reection coatings a periodic layer structure composed of dierentmaterials enhances the reectivity of the mirror because they lead the reected beams tointerfere constructively. By varying the thickness and the composition of the layers thecharacteristics of the mirror can be manipulated and high reective and anti-reective

38

4.3. Cavity Requirements

Table 4.1.: List of all wavelengths we want to stabilize.

Erbium DysprosiumWavelength (nm) Linewidth (kHz) Wavelength (nm) Linewidth (kHz)

583 190 626 136631 28 741 1.78

380–450 few kHz – –

regions can be created. Due to technical limitations, it is hard to produce one highlyreective (> 99.96 %) mirror coating for all our required wavelengths. Therefore, we usetwo sets of cavity mirrors of dierent mirror coatings. The mirrors for one cavity are coated1

for the 583 nm, 626 nm, 631 nm and 741 nm transitions, while the mirrors for the othercavity are coated for all wavelengths in the range from 380 nm–450 nm. Figure 4.4 showsthe theoretical transmission curves for the coatings, provided by Lens-Optics. Table 4.2gives the theoretical reectivity and the related values for F . For each wavelength thenesse for the theoretically best and worst case are calculated by Eq. 4.10. In comparison toother experiments, e. g. [4, 74] our cavities will have a relatively low nesse. Nevertheless,this is sucient since the light of the lasers does not have to be narrower than 1 kHz.

Table 4.2.: Reectivity and rough estimation of the expected nesse for the dierent wavelengths.For the calculations the theoretical curves from Fig. 4.4 and Eq. 4.10 are used. The 410 nm line is onepossible transition for the Rydberg excitation.

R(%) F

wavelength (nm) min max min max583 99.91 99.96 4500 7800

626 & 631 99.95 99.98 6300 20 000741 99.95 99.98 6300 15 700410 99.92 99.98 4000 15 700

Cavity StabilityTo improve the stability of the length of the cavity and therefore of the resonance fre-quency a spacer out of ultra low expansion glass2 (ULE) between the cavity mirrors isplaced. Although we have two cavities we use only one ULE spacer. This design is alreadyimplemented successfully in a running experiment [74]. The special property of ULE isthat the linear coecient of thermal expansion (CTE) has a zero-crossing point near roomtemperature providing a CTE3 below 2 × 10−9 1

K within one degree around this point and1The mirrors are manufactured and coated by Lens-Optics GmbH.2From Hellma-Optics, ULE premium grade.3Corning, manufacturer for glass and ceramic.

39

4. Design of the Vacuum Chamber and the ULE Cavity

(a)

(b)

Figure4.4.:The

dierentcoatings,usedforthe

two

setsofcavitym

irrorsareshow

n.(a)Coatingforthe

stabilizationofthe

583nm

,626nm

,631

nmand

741nm

lasers.(b)Coatingforlasersin

therangeof380nm

-450nm

usedforRydberg

excitation.Graphsareprovidedby

Lens-Optics.

40

4.3. Cavity Requirements

therefore high stability against length uctuations. Unfortunately the exact temperaturefor the zero-crossing can be dierent for each production series and is therefore speciedonly to be between 0 C and 30 C.

The cavity mirrors are optically contacted to the ULE spacer, which is manufacturedby Hellma-Optics. Also the optical contacting of the mirrors to the ULE spacer was doneby Hellma-Optics. Before choosing the mirror substrate we considered several aspects:Combining dierent production series of ULE or even dierent materials as ULE and FusedSilica (when using mirrors out of Fused Silica) could shift the zero-crossing temperatureout of the specied region. Fused Silica has a CTE of about 5 × 10−7 1

K being roughlyindependent of temperature. Due to the fact, that the CTE of Fused Silica is positive andthe CTE of ULE gets negative for temperatures below the zero-crossing, the minimum CTEof the ULE spacer together with the Fused Silica mirrors gets shifted to lower temperatures.This makes the temperature regulation more elaborate. Fused Silica mirrors have theadvantage that they can reduce the fractional instability from thermal noise comparedto ULE mirrors [51]. Nevertheless, for our purpose of a relatively low nesse cavity thestability of ULE mirrors does not limit us.

Beside the reectivity of the mirrors, it is important that the mirror substrate does notabsorb to much light. The transmission of ULE at our desired wavelengths is sucientlyhigh (see gray area in Fig. 4.5) and therefore, we decided to use cavity mirrors made out ofULE glass.

Figure 4.5.: Transmission curve of ULE glass. The gray area indicates approximately our wavelengthrange. Graphic taken from Präzisions Glas & Optik GmbH. December, 2016.

Regarding the long term frequency stability, we aim to stabilize the lasers in such away that they do not drift more than the linewidth of the transition over a long enoughtime period, ideally one day. In our case the smallest linewidth is 1.78 kHz for the 741 nmtransition of Dy. Thus, the cavity should have a long term stability on the order of ∆ν .1 kHz. The resonance frequency of a cavity depends on the optical length nL (see Eq. 4.7)and therefore the relative frequency deviation ∆ν of the cavity resonance can be expressed

41

4. Design of the Vacuum Chamber and the ULE Cavity

by the relative length change ∆L/L and the relative change of the refraction index ∆n/n:

∆ν

ν=

√(∆LL

)2

+

(∆nn

)2

. (4.17)

For our requirement of ∆ν . 1 kHz we need to reach a relative stability of

∆ν

ν≤ O(10−12) (4.18)

First, we look at the inuence of change in the refraction index: by using the Clausius-Mossotti Equation [52]

n2 − 1n2 + 2

=Nα3

(4.19)

we can make a connection between the refraction index n and both, the temperature Tand the pressure p. Since the refraction index for air is on the order of 1 + O

(10−4

)and

for vacuum even much lower, we can use the taylor expansion for Eq. 4.19, expanding n2

around 1, leading ton2 − 1n2 + 2

=n2 − 1

3.

Furthermore, inserting the molar refractivity A, which is a measure for the total polarizabilityof a mole, one obtains for the refractive index

n =

√1 +

3ApRT

, (4.20)

where R is the universal gas constant. For the relative change of the refractive index oneobtains

∆nn

=n − 1

n

√(∆pp

)2

+

(∆TT 2

)2

. (4.21)

Calculating the refractive index for a temperature and pressure comparable with our system,i. e. 20 °C and 10−8 mbar, leads to a relative change of refraction index on the order of 10−15.Obviously, the change in temperature and pressure should be moderate for this estimation.Therefore, in the following we can neglect the inuence of the refraction index on thestability of the cavity resonance and focus on the length stability.

The length change of the the ULE spacer can be described as

∆L = α(T ) · L · ∆T, (4.22)

whereα is the coecient of thermal expansion (CTE) and L the length of the cavity. A specialproperty of ULE is that the CTE has a so called zero-crossing around room temperature. InFigure 4.6 the CTE is plotted between 0 C and 44 C with the zero-crossing temperatureTC at 20 C.

This shows, that having a temperature stability of 10 C around TC gives a CTE on theorder of 1.5 × 10−8 /K. Assuming the CTE to be linear around TC gives a maximal CTE

42

4.3. Cavity Requirements

3

• CTE verification is achieved through a non-destructive ultrasonic method.• Stability: Excellent long term dimensional stability at room temperature. No residual figure change when taking a blank from 350 ˚C to water quench.• Delayed elastic effect: There has been no measurable delayed elastic effect in Corning 7973. This is an important consideration when large strain is present during fabrication or when environment loading is present, such as during gravity release or dynamic control of active optics. • No measurable hysteresis results from thermal cycling of Corning 7973.

Glass Code Striae Optical Retardation CTE Zero Cross Over Temperature

Coefficient of Thermal Expansion (CTE) Range

0 to 400 Scale[%]

Birefringence[nm/cm]

maximum

Tzc[˚C]

Radial[ppb/ ˚C]

Axial[ppb/ ˚C]

7973 Premium Grade 100 10 See note below* ≤ 10 ≤ 10

7973 Mirror Grade 100 20 See note below* ≤ 15 ≤ 15

7973 Standard Grade 100 20 See note below* ≤ 15 ≤ 15

7973 Tooling Grade NS NS See note below* ≤ 100 ≤ 100

7973 EUV Premium GradeCritical Zone: 50

10 User defined within 15 ˚C to 32 ˚C ± 5 °C ≤ 10 ≤ 10

Non-Critical Zone: 100

7973 EUV Standard GradeCritical Zone: 50

20 20 ºC ± 10 °C ≤ 15 ≤ 15Non-Critical Zone: 100

7973 Mask A Grade 50 NS 20 ºC ± 3 °C ≤ 6 N/A

Note:* Linear Coefficient of Thermal Expansion - The mean CTE shall be 0 ± 30 ppb/°C from 5 °C to 35 °C with a 95% confidence level and 0 ± 100 ppb/°C from 5 °C to 35 °C for Tooling Grade

Optical and Thermal Properties

Stress Optical Coefficient 4.15 (nm/cm)/(kg/cm2)

Striae Normal to Blank Faces None

Abbé Constant 53.1

D.C. Volume Resistivity, 200 ˚C 100 Hz (R) 1011.6 ohm•cm

Thermal Conductivity (K) 1.31 W/(m•K)

Unless otherwise stated, all values above @ 25 °C

Expansivity

-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.10.00.10.2

-150 -100 -50 0 50 100 150 200 250 300

ppm

/˚C

Temperature [˚C]

Instantaneous CTE

-20

0

20

40

60

80

100

-140 -120 -100 -80 -60 -40 -20 0 20 40 60 80 100 120 140

∆L/L

, ppm

Temperature [˚C]

Thermal Expansion

Thermal Diffusivity (D) 0.0079 cm2/s

Mean Specific Heat (Cp) 767 J/(kg•˚C)

Strain Point 890 ˚C

Annealing Point 1000 ˚C

Softening Point (estimated) 1490 ˚C

Unless otherwise stated, all values above @ 25 °C

-30.0

-20.0

-10.0

0.0

10.0

20.0

30.0

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44

Expa

nsivi

ty [p

pb/C

]

Temperature [˚C]

Instantaneous CTE

Temperature (˚C)

Expa

nsiv

ity (p

pb/˚C

)

0

10

-20

-30

-10

20

30

0 10 20 302 4 6 8 12 1814 16 322822 2624 34 36 38 40 42 44

Figure 4.6.: This Figure shows the zero-crossing of the instantaneous CTE for ULE at an exampletemperature of 20 C. Figure is taken from Corning, Product Information Sheet 7973 July, 2015

on the order of 1.5 × 10−11 /K for a temperature stability of 10 mK. Therefore, having atemperature stability on the order of 10 mK gives a relative length stability on the order of

∆LL

= O(10−11) · O(10−2) = O(10−13). (4.23)

This is easily sucient for our application (see Eq. 4.17).Another eect to take into account is the length change due to vibrations. The eect of

vibrations to relative length changes can be calculated by [15]

∆LL

=ρgh2E

σ, (4.24)

where ρ = 2.21 g/cm3 is the mass density, g is the acceleration, h the hight of the ULEspacer, E = 67.6 GPa the elastic modulus and σ = 0.17 the Poisson’s ratio4. This results ina sensitivity of the relative frequency stability to vertical accelerations of 2.8 × 10−10 /ms−2.Vibrations in the horizontal direction should have a much lower inuence because of thesymmetric mounting [45]. For the lab typical values for the vertical accelerations are onthe order of 3 mms−2, which leads to a minimum stability on the order of 10−13. Therefore,we can assume that the vibrational stability of the cavity is sucient for this purpose.

Another point to be noted is the drift of the frequency caused by relaxation eects ofthe cavity due to aging of the ULE spacer. In [26] the long term drifts of the cavity is ttedby two exponential decay functions superimposed on a linear function with a slope of13(2) mHz/s. The two time constants of the decay functions describing the long term driftsare 32(7) days and 260(80) days. The positive sign of the linear slope indicates that thecavity length decreases and is therefore an indication for aging of the material of the spacer.

4Values for the material specic parameters can be found into data sheet of ULE® Corning Code 7972.

43

4. Design of the Vacuum Chamber and the ULE Cavity

4.4. Mechanical and Electrical Design of the VacuumChamber

The design for the housing of the cavity was inspired by the master theses of AlexanderRitzler and Maximilian Segl [71, 74].

Vacuum Chamber Design

aluminum baseplate

CF63 viewportCF63 viewport

Sub-D25 feedthrough CF63

20 l/s ion-getter pumpedge-valve

Figure 4.7.: Drawing of the vacuum chamber which houses the ULE cavity. The baseplate has a sizeof 400 mm × 400 mm. The height of the chamber is about 300 mm. The vacuum chamber is made ofstainless steel, whereas all interior parts as well as the baseplate are made of aluminum. All vacuumparts are of the CF type. For optical access two CF63 viewports are implemented. A feedthroughwith a Sub-D25 connector is used for the connection of the NTCs and the Peltier elements and a 20l/s ion-getter pump is installed for pumping.

Fig. 4.7 shows the complete design. The vacuum chamber has a cylindrical shape and isclosed by two CF2505 anges. The vacuum chamber has two CF63 viewports for opticalaccess and an electrical feedthrough with a Sub-D25 connector for electrical connections.The viewports are attached at an angle of 3° to avoid disturbing reections. We use an

5ConFlat (CF) is a special ange design which uses a knife edge in combination with a copper gasket as asealing technique. The number stands for the DN type (DN250 in this case) and indicates the diameter ofthe system.

44

4.4. Mechanical and Electrical Design of the Vacuum Chamber

all-metal angled valve for the initial pump down with a turbo pump and a 20 l/s ion-pump6

to hold the pressure afterwards at the low 10−8 mbar level.The inner part of the vacuum chamber is formed by two consecutive shieldings as shown

in Fig. 4.8 which shows a cut through the vacuum chamber. The ULE spacer forming thecavity together with the mirrors is sitting on a V-shaped block supported by 4 rings madeof Viton, intended to reduce vibrational distortions. The location of the Viton rings isdetermined by the so called Airy points, a/L ≈ 0.2113, at which the bending of a held baris minimized and the facets of the end of the bar are parallel to each other. L relates to thelength of the ULE spacer and a is the distance between the Airy point and the nearer endof the bar.

We temperature stabilize the outer shielding via six Peltier elements7, whereas the innershielding is sitting on Teon spacers to ensure heat is transported mainly by radiation fromouter to inner shielding. The reduced heat transport results in a long thermalization timeconstant, which corresponds to the principle of a low-pass, i. e. fast temperature changesbelow the characteristic time constant are suppressed providing a higher stability againsttemperature uctuations. In Figure 4.9 the temperature of the aluminum block that holdsthe cavity is plotted against time. The temperature of the outer shielding was set from32.5 C to 30.5 C. With an exponential t the thermalization constant was determinedto 9.6(7) h. The dip around 25 h was probably caused by a temperature failure of the airconditioning system of the lab and is an artifact deriving from the measurement method.The temperature in the lab increased by a few degree and this inuenced the values of theresistors used for the voltage dividers to read out the temperature. The inner part of thecavity should not have experienced the temperature change.

Due to possible atness imperfections of the Peltier elements, we use tin sheets with athickness of 100 µm to compensate unevenness and therefore to increase the heat conduc-tivity. We glue coated windows into the holes of both shieldings to have optical access tothe cavities. Further, this should give additional shielding against thermal noise. All metallicinner parts of the vacuum chamber are made of aluminum for a better heat conductivity.To remove the heat produced by the Peltier elements, the vacuum chamber sits on a largeheat sink. To reduce the temperature dierence for the Pelter elements that stabilize theouter shielding six Peltier elements are used to cool the lower ange of the cavity. At thesame time this Peltier elements temperature stabilize the vacuum chamber. At the timethe housing for the cavity was built the CF250 anges and the tube were available onlyout of stainless steal and therefore the vacuum chamber itself has a 5 times less thermalconductance than aluminum. Therefore, a copper plate sitting on the Peltier elements wasincluded to increase the contact area from the Peltier elements on the baseplate to thelower ange of the vacuum. For future cavity projects aluminum CF vacuum componentsare recommended. Figure 4.10 shows temperature simulations8 for both cases, with andwithout the additional copper plate. If we have to go to low temperatures (CTE of ULEis specied between 0 °C and 30 °C) the heat sink is not big enough to remove the heat

6Agilent Vaclon Plus 20.7Multicomb MCPE-241-10-13 Peltier Cooler, 71.8 W.8SolidWorks 2014.

45

4. Design of the Vacuum Chamber and the ULE Cavity

inner shieldingouter shielding

ULE cavity

copper block for better thermal connection

3˚

NTC

aluminum heat sink

Teflon spacerPeltier element

(a) Cut through the vacuum chamber. The cavity spacer is seen from the side.

583nm, 626nm, 631nm, 741nm

380nm - 450nm

(b) Cut through the vacuum chamber. The cavity spacer is seen from the front.

Figure 4.8.: Vertical cuts through the vacuum chamber. In (a) the cavity spacer is seen from theside. The Peltier elements are illustrated in green and the Teon spacer in red. The position of theNTCs which are used for temperature regulation are marked by a black dot. Many other NTCs aredistributed over the vacuum chamber. In (b) the cavity spacer is seen from the front and the twocavities with the dierent coatings are labeled.

46

4.4. Mechanical and Electrical Design of the Vacuum Chamber

time (h)0 5 10 15 20 25 30 35 40 45 50

30

30.5

31

31.5

32

32.5

33

Figure 4.9.: Record of the temperature of the aluminum holder of the cavity. Note that the dip at25 h was caused by a failure of the air conditioning system of the lab. The data points are ttedwith an exponential function y = A + B exp(t/τ), where t denotes the time. The time constant τ wasdetermined to 9.6(7) h.

produced by the Peltier elements. The heat transport of the vacuum chamber to the outershielding by thermal radiation can be calculated by

Q = εσ(AchamberT 4

chamber − AshieldingT 4shielding

)(4.25)

where ε is the emissivity of the material, σ is the Stefan-Boltzmann constant, A the surfacearea and T is the temperature. By assuming 0.2 for ε and a temperature dierence of 10 Cleads to a thermal radiation of 15 W from the vacuum chamber to the shielding. This meansthat 15 W of the cooling power of the Peltier elements are used to make up the heatingfrom the vacuum chamber walls.

The temperature dierence between the shielding and the vacuum chamber should beas low as possible and therefore a water cooling system is implemented in the baseplate.The water cooling system consists of a copper tube that was pressed into a channel on thebottom of the baseplate. Testing the reachable temperature with the water cooling showedthat this simple solution is satisfying for our purpose.

Temperature ControlThermistors9 (NTCs) for the temperature regulation and monitoring are allocated at dierentpositions in the vacuum chamber. The monitoring of the NTCs is done with an Arduinosystem10. The two NTCs planned for the temperature regulation are sitting on the copperplate below the lower ange of the vacuum chamber and on the outer shielding (see Fig. 4.8).The temperatures are stabilized with a simple analog PID controller. In the Appendix A.3,

9Epcos B57541G1, 10 kΩ at 25 C.10Arduino Due.

47

4. Design of the Vacuum Chamber and the ULE Cavity

the measurement bridge, which generates the error signal, and the PID drawing are shown,respectively. For the connection of the electronic components inside the vacuum chamberwe use wires insulated with Kapton™11. The advantage of using Kapton™is that it has alow outgassing rate compared to standard isolation materials. The wires are soldered tothe Sub-D connector at the feedthrough and the electronic components and to isolate thesolder joints we use tape out of Kapton™.

11Twisted pair wire, 311-KAPM-060-PAIR1-20M.

48

4.4. Mechanical and Electrical Design of the Vacuum Chamber

(a) Heat sink and lower ange without the additional copper plate.

(b) Heat sink and lower ange with the additional copper plate.

Figure 4.10.: Temperature simulations, done with SolidWorks 2014, for the two types of base plateswe tested. It is assumed, that six Peltier elements are sitting on the lower ange, which are heatingwith a total power of 100 W. Below the lower ange 6 Peltier elements should remove the heat andtransport it to the aluminium heat sink. In (a) no additional copper plate was placed and the heatcan not be removed satisfyingly. In (b) the copper plate solves the issue of the heat transport.

49

4. Design of the Vacuum Chamber and the ULE Cavity

50

5. Experimental PartIn this Chapter, we present the main experimental results concerning the characterizationof the cavity performance, i. e. the zero-crossing temperature of the CTE of the ULE glass,the nesse, and the free spectral range of the cavity.

5.1. Optical Setup for the Coupling of the Lasers to theCavity

For a good mode matching between the light and the cavity, it is important to fulll thecondition from Eq. 4.16

w0 =Lλπ

√ρ − L

L, (5.1)

where λ is the wavelength, L is the length of the cavity and ρ is the radius of the curvedmirror. Table 5.1 gives the theoretically calculated values for the necessary beam waistson the planar mirror wplanar of the cavity. These beam waists can be achieved, by putting a

Table 5.1.: Calculated beam waists on the planar mirror of the cavity wplanar for the light of 583 nm,626 nm, 631 nm, and 741 nm. The cavity length is L = 15 cm and the radius of the curved mirror isρ = 500 mm.

λ (nm) wplanar (mm)583 0.206626 0.2137631 0.2145741 0.2325

lens in front of the cavity to focus the beam. Figure 5.1 shows the lens in front of the cavity,forming a lens system with the curved mirror of the cavity. We assume the curved mirrorwith ρ = −500 mm being a lens with f =

ρ

n−1 = −1000 mm, because the refractive index nof ULE is approximately 1.51.

We calculate the beam waist on the planar mirror of the cavity in dependence of thewaist of the incoming collimated beam via ray transfer matrix analysis. For a lens with afocal length of 500 mm the incoming beam should have a beam waist of ≈ 0.5 mm. Ourber output coupler2 is chosen such that the beam diameter is already on the right order

1Corning, manufacturer for glass and ceramic.2Thorlabs, SM1Z in combination with CP11/M.

51

5. Experimental Part

15cm35cm

ρ = 500 mmf = 500 mm

incoming beam

windows inner shielding outer shielding

Figure 5.1.: Schematic presentation for the coupling to the cavity. The laser beam gets focusedthrough the 500 mm lens into the cavity, whereby the curved mirror can be assumed as a lens withfocal length −1000 mm.

and for ne adjustment the position of the lens and therefore the beam size can be tuned.Figure 5.2 shows the optical setup for coupling of the 626-nm light and the 583-nm lightinto the cavity.

EOM

lensf = 50mm

PD

PBS PBSHCL

λ/2

λ/2

λ/2

fiber outputcoupler

mixerLO 5 MHz

EO

M

EO

M

PDPD

lensf = 50mm

lensf = 50mm

BS

BS

fiber outputcoupler

fiber outputcoupler

lensf = 500mm

M1 M1

M2

M2

dichroic mirror

EO

M

PD

PID

Laser

BS

PBSexperiment

λ/2

cavity

cavity

fmod + fshiftfmod

583 nm

626 nm

1050

nm

1550

nm

PPN

L

lensf = 50mm

lensf = 100mm

lensf = 200mmlensf = 50mm

lensf = 100mm

λ/2λ/4

λ/4λ/2 lens

f = 200mm

lensf = 150mm

λ/2

λ/2

PBS

PBS

λ/2

experiment

wavemeter

cavity

spectroscopy signal

Figure 5.2.: Optical setup to couple the light of 583 nm and 626 nm into the cavity. After the beroutput coupler the light gets coupled into the cavity with the mirrors M1 and M2. A dichroic mirroroverlaps the 583-nm light (yellow) with the 626-nm light (orange). The light is then focused ontothe planar mirror. The reected light is picked up with a beam splitter (BS) and detected with aphotodiode (PD).

52

5.2. Modulation-Transfer Spectroscopy

We guide the laser beams via the ber-coupled EOM to a breadboard, where the setup tocouple the light into the cavity is placed. A dichroic mirror3 overlaps the 583-nm and the626-nm light and we couple the light with the mirrors labeled by M1 and M2 into the cavity.We focus both beams into the cavity through the same lens. After the output coupler a50-50 beam splitter splits the reected signal of the cavity for the generation of the PDHerror signal. One reason to use a 50-50 beam splitter instead of a polarizing beam splitter incombination with a λ/4 in front of the cavity, as in [71], is that we want to add the 631-nmlight and the 741-nm light in the future and there is no standard dichroic mirror available tooverlap the 626-nm and the 631-nm light. Therefore, the light has to be overlapped with aPBS and with a λ/4 in front of the cavity the separation of the reected signal is not possible.

5.2. Modulation-Transfer SpectroscopyTo achieve a stable operation of the cavity, one rst needs to determine the value of thetemperature at which the cavity, i. e. ULE mirrors and ULE spacer, exhibits a minimalthermal expansion. To determine this temperature, we measure the resonance frequency ofthe cavity because a change of the cavity length results in a change of the free spectral range(FSR) (see Sec. 5.3). We measure the relative length of the cavity by measuring changes in theresonance frequency with respect to the frequency of a reference signal. At the time whenthis measurement was done we had only the laser of 626 nm for Dy available and thereforewe used this laser for the measurement. Since Dy has a high melting point (see Sec. 2.1),one would need a high temperature oven to create neutral hot Dy vapor. Instead, we use ahollow cathode lamp (HCL) to produce the atomic sample. The HCL is a glass cell with acathode and an anode inside. The cathode is coated with the desired element, in our casewith dysprosium. The glass cell is lled with argon, which serves as a buer gas and getsionized in an electric discharge process when a high voltage is applied between cathode andanode. The applied voltage during the spectroscopy was 117.1 V with a current of 14.9 mA.After the ionization the Ar+ ions are accelerated and impact into the cathode. In this waythe Ar+ ions sputter the dysprosium away in form of atomic vapor by depositing theirkinetic energy. The hot atomic vapor exhibits a too large Doppler broadening to be used asa reference. In order to avoid this we use a Doppler-free spectroscopy technique, namelythe modulation-transfer spectroscopy. This method employ two counter-propagating laserbeams to interrogate the atoms. One beam, known as pump beam, has a higher intensitywith respect to the weak probe beam. Figure 5.3 shows the setup for the generation of theMTS signal and Appendix A.2 outlines the generation of the spectroscopy signal.

We guide the laser light with a ber from the setup of the 626-nm laser to the spectroscopysetup. After the ber output coupler the light is divided into pump and probe beam with aλ2 -waveplate and a polarizing beam splitter (PBS). The best signal-to-noise ratio has beenobtained using a relative power ratio of approximately Ipump/Iprobe ≈ 4. The pump beam goesthrough an EOM, which imprints sidebands onto the beam at a modulation frequency of5 MHz. The frequency is generated by a direct-digital synthesizer (DDS) and enhancedby an LC-circuit, formed by the EOM crystal that acts as a capacitor and an inductance

3Thorlabs, DMLP605.

53

5. Experimental Part

EOM

lensf = 50mm

PD

PBS PBSHCL

λ/2

λ/2

λ/2

fiber outputcoupler

mixerLO 5 MHz

EO

M

EO

M

PDPD

lensf = 50mm

lensf = 50mm

BS

BS

fiber outputcoupler

fiber outputcoupler

lensf = 500mm

M1 M1

M2

M2

dichroic mirror

EO

M

PD

PID

Laser

BS

PBSexperiment

λ/2

cavity

cavity

fmod + fshiftfmod

583 nm

626 nm

1050

nm

1550

nm

PPN

L

lensf = 50mm

lensf = 100mm

lensf = 200mmlensf = 50mm

lensf = 100mm

λ/2λ/4

λ/4λ/2 lens

f = 200mm

lensf = 150mm

λ/2

λ/2

PBS

PBS

λ/2

experiment

wavemeter

cavity

spectroscopy signal

Figure 5.3.: The setup for the MTS. The ratio between the pump beam and the probe beam isIpump/Iprobe ≈ 4. The modulation frequency applied to the EOM is 5 MHz and generated by a localoscillator (LO), which consists of a frequency generator in combination with a LC-circuit. The probebeam is detected with a photodiode. To obtain the nal spectroscopy signal, the photodiode signalis mixed with the modulation frequency of the EOM. PD: photodiode, PBS: polarizing beam splitter,HCL: hollow cathode lamp.

of L = 56 µH. Afterwards, the pump beam gets overlapped with the probe beam in the HCL.Figure 5.4 shows an example of the MTS signal for Dy. Two transitions can be assignedto the bosonic isotopes 164Dy and 162Dy, whereas three transitions can be assigned tofermionic isotopes. The amplitude of the resonances for the bosonic isotopes comparedto the fermionic isotopes is much higher because the additional hyperne structure ofthe fermions leads to more states, i. e. 9 for Dy, over which the fermions are distributed.Therefore, the number of fermions resonant to the respective light is only slightly morethan 10 % of all fermions. The resonance of the bosonic 160Dy and the resonances of theother fermionic transitions are out of the scan range.

The frequency dierence between the 164Dy and the 162Dy resonance can be extractedby tting the theoretical spectroscopy signal, Eq. 5.9, to the data and is determined to be960.2(1) MHz. This agrees within the error bars with the value of 962(2) MHz obtainedin [41]. The transition frequencies of the hyperne states relative to the bosonic isotope162Dy can be calculated by

∆νHFS = ∆162 + Ee(Fe, Je, I) − Eg(Fg, Jg, I), (5.2)

where ∆162 is the isotope shift with respect to the bosonic 162Dy and Ei are the hyperneenergies

Ei =12

AiCi +12

Bi3Ci (Ci + 1) − 4I (I + 1) Ji (Ji + 1)

2I (2I − 1) 2Ji (2Ji − 1)(5.3)

withCi = Fi (Fi + 1) − Ji (Ji + 1) − I (I + 1) . (5.4)

54

5.2. Modulation-Transfer Spectroscopy

frequency relative to the 162Dy transition (GHz)-1.7 -1.5 -1.3 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1 0.1 0.3

MT

S si

gnal

(m

V)

-60

-40

-20

0

20

40

60

162Dy164Dy

163DyF = 17

2! F = 19

2

161DyF = 21

2! F = 23

2

163DyF = 15

2! F = 17

2

Figure 5.4.: Modulation-transfer spectroscopy signal for dysprosium. Two transitions can beassigned to the bosonic isotopes, whereas three transition are of fermionic isotopes. The hypernetransitions are labeled by F → F′. The signal was averaged over ten recording cycles.

Table 5.2 gives the hyperne coecients A and B and Table 5.3 gives the calculated andmeasured frequency dierences of the hyperne transitions relative to the 162Dy bosonicisotope that corresponds to Fig. 5.4. The measured frequency dierences agree within theerror bars with the theoretically calculated ones.

Table 5.2.: Hyperne A and B coecients for the ground state (g. s.) and the excited state (e. s.) ofthe 626-nm transition [41], [29] (as cited in [58]).

g. s. A (MHz) B (MHz)161Dy −116.2322 1091.574163Dy 162.7543 1153.8684e. s.

161Dy −187.0(1) −317(4)163Dy 261.7(2) −334(6)

In principle, already the spectroscopy signal could be used to lock the laser. However, forthe 583-nm transition of Er and for the 626-nm transition of Dy broadening eects suchas power broadening and pressure broadening aect the spectroscopy signal. The powerbroadening can be calculated by [31]

∆ωFWHM = 2π · ∆ν(1 +

IIsat

)1/2

, (5.5)

where ∆ν is the natural linewidth and Isat is the saturation intensity of the transition. ForDy a power of 4 mW leads to a power broadening of ∆ωFWHM ≥ 2π · 1 MHz. Additionally,

55

5. Experimental Part

Table 5.3.: Calculated hyperne transitions relative to the bosonic 162Dy. For the calculations Eq. 5.3is used and Table 5.2 gives the hyperne coecients A and B.

transition ∆ν (MHz) ∆νmeasured (MHz)163Dy: F = 17/2→ F′ = 19/2 212(2) 210(8)163Dy: F = 15/2→ F′ = 17/2 −1014(2) −1011.0(1)161Dy: F = 21/2→ F′ = 23/2 −1470(2) −1473.0(1)

the high background pressure in the HCL of 4 mbar, leads to pressure broadening of thetransitions. The pressure broadening can be as large as 8.2 MHz [32]. Due to this twobroadening eects the signal gets very large and therefore, this method is not optimalto lock the light of 583 nm and 626 nm, because the linewidths of these transitions areΓ = 2π · 190 kHz and Γ = 2π · 136 kHz, respectively. By tting the data points of ourspectroscopy signal we extract a linewidth of Γ = 2π · ∆ν = 12.051(5) MHz for a totalpower of the two beams of approximately 10 mW. In Figure 5.5 one clearly sees, that thetransmission peaks of the cavity are much narrower that the MTS spectroscopy signal andthus better for the locking purpose.

The spectroscopy still gives an absolute frequency reference to calibrate the frequencyof the cavity modes by comparing the spectroscopy signal with the cavity transmission.Furthermore, it gives a stable reference, which can be used to determine the zero-crossingof the CTE of the cavity as discussed in the following section.

5.3. Zero Crossing of the ULE CavityOne of the advantages of using mirrors and a cavity spacer out of ULE is that the coecientof thermal expansion (CTE), α of ULE has a zero-crossing around room temperature (seeSec. 4.3 and Fig. 4.6). If the cavity is temperature stabilized at this special zero-crossingtemperature TC the length of the cavity becomes more stable against temperature uctua-tions. For cavities stabilized at TC frequency drifts on the order of Hz/s due to temperatureuctuations can be achieved [46, 71].

The cavity mirrors are optically contacted to the spacer and since both elements aremade of ULE we expect TC to be between 0 C and 30 C.

Around TC the CTE of ULE can be approximated by

α(T ) = a (T − TC) + b (T − TC)2 . (5.6)

The linear expansion coecient a is on the order of 10−9 /K2 and the quadratic expansioncoecient b is on the order of 10−12 /K2 [51]. Since we are interested only in a small regionaround TC, keeping (T − TC) small, and b is three orders smaller than a, the linear term isdominant (compare to Fig. 4.6) and we focus only on the linear approximation, which leadsto a length change of

dL = Lα (T ) dT = La (T − TC) dT. (5.7)

56

5.3. Zero Crossing of the ULE Cavity

The relative change in frequency is directly connected to the change in length by

∆ν

ν=

∆LL

=a2

(T − TC)2 (5.8)

To determine TC, we measure the frequency drift of the cavity resonance while changing thetemperature of the cavity. The frequency drift is quantied by measuring the position of thecavity transmission relative to the modulation-transfer spectroscopy signal of the 626-nmtransition for 162Dy. For this the laser is scanned over more than one FSR and the respectivetraces are recorded with an oscilloscope4. Figure 5.5 shows the cavity transmission signaland the spectroscopy signal. The cavity transmission signal is tted with a double Lorentzianfunction, whereas the spectroscopy signal is tted with the theoretical spectroscopy signalderived in [63]

S (ωm) =C√

Γ2 + ω2m

×[(

L−1 − L−1/2 + L1/2 − L1)

cos(φ)

+(D1 − D1/2 − D−1/2 + D−1

)sin

(φ)],

(5.9)

whereLn =

Γ2

Γ2 + (∆ − nωm)2 (5.10)

andDn =

Γ(∆ − nωm)Γ2 + (∆ − nωm)2 . (5.11)

Γ is the linewidth of the transition, ωm the modulation frequency, ∆ the detuning fromthe atomic resonance and φ is the phase dierence between the detected signal and themodulation eld applied to the EOM to modulate the pump beam. The center position ofthe transmission peak of the cavity labeled with T in Fig. 5.5 is taken as frequency reference.

Additionally, we record the voltage at an NTC sitting on the aluminum block that holdsthe cavity (see Sec. 4.4) and use the associated temperature as actual cavity temperature.The NTC is part of a voltage divider and the temperature can be read out by calculating theresistance of the NTC. We changed the temperature from 27.0 C to 17 C in steps of 2 C.Since the cavity has no thermal connection to the aluminum block, where the temperatureis measured, the time between setting the temperature and measuring the relative positionof the resonance was minimum 2 days. This ensures that the cavity has thermalized andreached the same temperature as the aluminum block. After reaching 17 C we changedthe temperature back to higher values to check if the time we waited for thermalizationwas enough or if we observe a hysteresis behavior as in [71]. Figure 5.6 shows the positionof the 162Dy transition relative to the cavity transmission as a function of the temperatureof the aluminum block that holds the cavity. We do not observe a hysteresis behavior,which means, that the time we waited for thermalization was enough. For each data point

4LeCroy: SDA 760Zi-A.

57

5. Experimental Part

frequency relative to the cavity transmission (GHz)-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2

PD /

MT

S si

gnal

(m

V)

-60

-40

-20

0

20

40

60162Dy T

"8

Figure 5.5.: Example of the spectroscopic trace recorded with the oscilloscope for the determinationof the zero-crossing. The aluminum block that holds the cavity had a temperature of 16.58(6) C

approximately 50 traces were recorded and the position was averaged. The error bars inFig. 5.6 are the standard deviation, whereby the error bars for the temperature are not visible,because they are on the order of 5 × 10−2 C. The data points are tted with a quadraticfunction ∆ν = c + νa(T − TC)2, where c is a constant containing the minimal dierencebetween the atomic resonance and the cavity transmission. From the t we extract TC to be

TC = 21.78(5) C.

and the linear expansion coecient a in Eq. 5.6 to be

a = 1.65(5) × 10−9 /K.

The error is the statistical error from the t. We determine the temperature of the zero-crossing of the CTE with a precision of 50 mK, which satises the requirements givenin Sec. 4.3. The result of 21.78(5) C for TC is very convenient because the temperaturestabilization is very easy, i. e. no water cooling is required. Additionally, the dierence tothe room temperature is small and therefore the Peltier elements do not have to work athigh power. In the following the cavity is stabilized at 21.78(5) C and the locking of the626 nm lasers is done.

5.4. Locking of the Laser SystemsThe Pound-Drever-Hall (PDH) technique (see App. A.1) is used to lock the lasers for thelight of 583 nm and of 626 nm. The idea is to modulate the laser light via a ber-coupledEOM5 with two dierent modulation frequencies, which results in sidebands at frequenciesfshift and fmod. The frequencies are generated by two direct-digital synthesizer (DDS) andsuperimposed with a power combiner6. While the laser frequency is tuned to the atomic

5Jenoptik, PM635 (PM585) for the light of 626 nm (583 nm).6Minicircuits, ZSC-2-4+.

58

5.4. Locking of the Laser Systems

15 20 25 30

posi

tion "8 (

MH

z)

-145

-135

-125

-115

Figure 5.6.: The frequency dierence of the 162Dy spectroscopy transition and the cavity resonanceis plotted as a function of the temperature of the aluminum block that hold the cavity. The datapoints are tted with a quadratic function ∆ν = c + νa(T − TC)2. Each point is averaged over about50 data points. The error bars are given by the standard deviation. The zero-crossing temperature isat 21.78(5) °C and a = 1.65(5) × 10−9 /K.

resonance, the locking is done with the additional sidebands by shifting fshift to the cavityresonance and using this sideband together with the sidebands at fmod to generate the errorsignal. In Figure 5.7 the locking scheme is illustrated schematically. fexp labels the atomicresonance frequency and fn labels the cavity modes. Figure 5.8 shows the schematic setup

fn fn+1fexp fn fn+1fexp

EOM

FSR fshiftfmod

Figure 5.7.: Illustration of the locking scheme for the laser of light of 626 nm and 583 nm. fn labelsthe cavity modes and fexp labels the experimentally desired wavelength. fshift and fmod are thesidebands that are modulated by the EOM.

for the laser locking. The light of the laser gets divided by a polarizing beam splitter (PBS).Most of the power goes to the experiment and only approximately 5 mW are used for thelaser locking. A ber-coupled EOM modulates the sidebands at frequencies fshift and fmod.The reected beam of the cavity is intercepted with a beam splitter and recorded with aphotodiode. The detected signal is mixed with the modulation frequency fmod to extract thePDH error signal. The laser is then stabilized with a PID controller. This locking methodhas several advantages compared to other systems realized with acousto-optic modulators(AOM) as in [58, 71]: The ber-coupled EOM has a large bandwidth of 5 GHz, which gives

59

5. Experimental Part

EOM

lensf = 50mm

PD

PBS PBSHCL

λ/2

λ/2

λ/2

fiber outputcoupler

mixerLO 5 MHz

EO

M

EO

M

PDPD

lensf = 50mm

lensf = 50mm

BS

BS

fiber outputcoupler

fiber outputcoupler

lensf = 500mm

M1 M1

M2

M2

dichroic mirror

PD

PID

Laser

BS

PBSexperiment

λ/2

cavity

cavity

fmod + fshiftfmod

583 nm

626 nm

1050

nm

1550

nm

PPN

L

f = 50mmf = 100mm

f = 200mm

f = 50mm

f = 100mmλ/2λ/4

λ/4λ/2

f = 200mm

f = 150mm

λ/2

λ/2

PBS

PBS

λ/2

experiment

wavemeter

cavity

spectroscopy signal

ECDL

λ/2

λ/2

PBS

AOM

SHG

λ/2 λ/2

λ/2PBS

PBS

AOM

fiber laser+

fiber amplifier

1166 nm

583 nm

wavemeter

cavity wavemeterwavemeter

experiment

f = 175mm

f = -50 mm

f = 200 mm

f = 100 mm

f = -50 mm

f = 50 mm

f = -30 mm

optical isolatorE

OM

Figure 5.8.: Schematic drawing of the setup of the laser locking for the 626 nm light.

a large tuning range (large fshift is possible) without the need to optimize the couplings tothe AOMs. This facilitates the switching between the dierent isotopes. The optical setupgets much more easier compared to standard realizations of such systems [32, 58] becauseusually at least two AOMs in double pass conguration are necessary to obtain a sucientlylarge frequency scan range for the laser light. The ber-coupled EOM also provides a spatialmode cleaning, which makes the coupling into the cavity and the suppression of highertransversal modes easier. In total only one ber-coupled EOM and one AOM are requiredwith our system (see Fig. 3.7 and 5.8) for tuning, locking and power stabilization of the laserlight. Figure 5.9 shows the PDH error signal that we obtain with a modulation frequencyfmod of 20 MHz. In Section 5.5 we extract the linewidth and therefore the nesse out of theerror signal. Figure 5.10 shows the cavity transmission signal and the signal of the MTS,while the laser was scanned over more than one FSR. The sidebands are at a frequency of200 MHz, whereas the sidebands at 20 MHz used for locking are not visible due to the lowamplitude. This plot can be used as a reference to check how large the frequency fshift mustbe, to shift the sideband to the cavity resonance to lock the laser light of 626 nm.

5.5. Performance of the CavityBeside the zero-crossing of the CTE, the free spectral range (FSR), the nesse, and thelinewidth of the cavity are important parameters. In the following section we determinethis three properties with dierent measurement methods. For all three measurements weuse the light of 626 nm.

60

5.5. Performance of the Cavity

frequency (MHz)-30 -20 -10 0 10 20 30

volta

ge (

mV

)

-800

-600

-400

-200

0

200

400

600

Figure 5.9.: PDH error signal of the 626 nm light obtained with the cavity.

Free Spectral Range and Linewidth out of the Transmission SignalIn this subsection we extract the FSR and the linewidth out of the transmission signal of thecavity. For this, we modulate sidebands on the laser light with the ber-coupled EOM andscan the frequency of the laser over more than one FSR (1 GHz). We record the transmissionof the cavity with an oscilloscope7 and use the sidebands at known frequencies to convertthe time axis into a frequency axis. Figure 5.10 shows the transmission of the cavity withsidebands modulated at 200 MHz. To measure the FSR we take the frequency dierencebetween the center positions of the carrier of the transmission peaks (labled with C inFig. 5.10), which are tted with a Lorentzian function

f (x) =1π

As2 + (x − t)2 . (5.12)

Figure 5.11 shows a zoom into one cavity transmission peak and the Lorentzian t. Wedid this with three dierent modulation frequencies, 100 MHz, 120 MHz, and 200 MHz andaverage the FSR to

F = 989(8) MHz.

The slightly lower value than the expected FSR of 1 GHz can be explained by the opticallycontacted mirrors. Due to the mirrors, especially the curved mirror, the cavity distance isslightly increased, leading to a smaller free spectral range.

Since the linewidth of our 626 nm laser light is on the order of a few 10 kHz and thereforelower than the linewidth of the cavity for the theoretically best coating case of the mirrors(≈ 50 kHz), it is possible to measure the linewidth by scanning the laser over the cavityresonance. We t each transmission peak in Fig. 5.10 with the Lorentzian Function 5.12 andextract the linewidth. All 6 linewidths are then averaged to ∆ν = 630(9) kHz.

7LeCroy: SDA 760Zi-A.

61

5. Experimental Part

frequency relative to the cavity transmission (GHz)-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4

PDvoltage(mV)

-20

-10

0

10

20

30

40

50

162Dy164Dy

163DyF = 17

2! F 0 = 19

2

161DyF = 21

2! F 0 = 23

2

163DyF = 15

2! F 0 = 17

2

CC

Figure 5.10.: Cavity transmission and the MTS signal. The laser is scanned over more than onefree spectral range with a modulation frequency of 200 MHz switched on. The 20 MHz sidebands,that will be used for the generation of the PDH-signal, are not visible here because there amplitudeis too small.

By combining the results obtained for the FSR and the linewidth of the cavity we calculatethe nesse with Eq. 4.9 to

F = 1569(23). (5.13)

Finesse and Linewidth out of the Cavity Ring-Down MeasurementAnother method to determine the linewidth of the cavity and to calculate the nesse withEq. 4.9 is the cavity ring-down measurement. The working principle of the cavity ring-downmeasurement is based on the decay time of the light stored in the cavity. If the laser islocked to the cavity light enters the cavity and due to the reections of the mirrors a part ofthe light remains trapped in the cavity modes. The equilibrium between the leaving Iout(t)and entering intensity Iin(t) is described by

Iout(t) =1τ

(Iin(t) − Iout(t)) . (5.14)

τ is the decay constant and dened as

τ =1αc, (5.15)

where α is the total lost per length unit and c the speed of light. If at t = ξ, the incomingintensity is suddenly switch of, Iin(ξ) = 0, then Eq. 5.14 reduces to

Iout(t > ξ) = −1τ

Iout(t). (5.16)

62

5.5. Performance of the Cavity

frequency (kHz)-1000 -800 -600 -400 -200 0 200 400 600 800 1000

PD v

olta

ge (

mV

)

15

20

25

30

35

40

FWHM

Figure 5.11.: Zoom into one cavity transmission peak. The cavity resonance is tted with theLorentz function from Eq. 5.12. The frequency axis is set to be zero at the resonance.

This dierential equation has the simple solution

Iout = I0e−t/τ, (5.17)

where I0 is the intensity stored in the cavity. For this measurement we lock the 626-nmlaser onto the cavity resonance as explained in Sec. 5.4 and change suddenly the frequencyfshift of the sideband, which is shifted to the cavity resonance. Since the sideband is notresonant to the cavity anymore, the input intensity is turned o. The intensity transmittedthrough the cavity is recorded with a photodiode. Figure 5.12 shows the exponential decayof the measured intensity. It is important that the switching occurs much faster than thedecay time. Otherwise, the assumption of Iin(t > ξ) = 0 is not valid anymore and the decayis determined by the switching time. The EOM has a specied switching time of 0.2 ns andis therefore fast enough for this purpose, since we expect a decay time τ > 100 ns.

To extract the decay time, the exponential function

f (t) = A + B · e−tτ (5.18)

is tted to the data. The constant A describes the intensity osets, including stray light andthe oset from the photodiode and B is the amplitude of the decay function. The photonlifetime can be determined to

τ = 260.4(7) ns,

which results, in a linewidth of the cavity of

∆ν = 611(2) kHz,

63

5. Experimental Part

time (7s)-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

PD

voltage

(mV

)

-5

0

5

10

15

20

25

30

35

Figure 5.12.: Cavity transmission signal as a function of time. The intensity decays after shiftingthe sideband, on which the laser is locked, out of resonance.

by using Eq. 4.11. Combining the FSR of 989(8) MHz calculated with the cavity transmissionand the obtained linewidth, the nesse of the cavity can be calculated using Eq. 4.9 to

F = 1636(4).

Linewidth out of the PDH Error SignalBeside the determination of the linewidth out of the cavity transmission signal and thecavity ring-down as a third method the linewidth is extracted out of the PDH error signal.This is done, by tting [62]

S (∆) = A + B ·Γ∆Ω

(sin(ϕ)Γ

(Γ2 + ∆2 + Ω2

)+ cos(ϕ)Ω

(Γ2 − ∆2 + Ω2

))(Γ2 + ∆2) (Γ2 + (∆ + Ω)2

) (Γ2 + (∆ −Ω)2

) (5.19)

to the data points, where A is a general oset, B is the amplitude of the signal, Γ is thelinewidth of the cavity, ∆ is the laser detuning with respect to the cavity resonance, Ω is themodulation frequency and ϕ is the phase between the reected signal and the modulationsignal.

In Figure 5.9 shows the t to the recorded error signal. We extract a linewidth of410(20) kHz. By using the FSR of 989(8) obtained before we can calculate the nesseto

F = 2213(120). (5.20)Function 5.19 does not t perfect to the data. This explains the deviation from the resultsobtained with the cavity transmission and the ring-down measurement. Nevertheless, thelinewidth and therefore the nesse is of the same order and conrms the other results.

64

5.5. Performance of the Cavity

Discussion of the Cavity ParametersTable 5.4 gives a summary of the results obtained via the three dierent measurementmethods discussed above. Theoretically, following the coating curves for the mirrors inFig. 4.4 provided by Lens-Optics8 and using Eq. 4.10, one obtains for the nesse for light at626 nm a minimum value of 6300, which is approximately a factor of 4 higher than in ourcase.

Table 5.4.: Summary of the results obtained for the nesse F , the FSR, the linewidth ∆ν, and thereection coecient R with the dierent measurement methods.

min. theo. calculated trans. signal cavity ring-down PDH error signalFSR (MHZ) 1000 989(8) – –F 6300 1569(23) 1636(4) 2213(120)

∆ν (kHZ) 159 630(9) 611(2) 410(20)R (%) 99.95 99.800(3) 99.8082(5) 99.86(1)

One reason for the low nesse could be that the mirrors are not as good as specied.To verify the quality of the mirrors we measured the transmission for both type of mirrorcoatings with two mirrors, which we have in spare. We used light of 421 nm for the mirrorscoated from 380 nm – 450 nm and light of 626 nm for the other mirror coating type. Table 5.5gives the reection coecients that are calculated with the measured transmission. Totake into account that some of the light is absorbed from the mirror substrate we assumean absorption of 10 %, according to Fig. 4.5. Although the absorption is estimated quitehigh, the nesse calculated for 626-nm with the measured transmission coecient lieswithin the specications and is higher than the nesses measured with the cavity. As acheck we estimate the nesse of the cavity also for the light at 421 nm. For this we recordthe transmission with the oscilloscope and estimated the linewidth to be > 1 MHz. Thecorresponding nesse is lower than 1000. This leads to the conclusion that the mirrorcoatings are not casing the low nesse.

Table 5.5.: Measured transmission for both type of mirror coatings. Additionally the resultingnesse is calculated with this reection coecient and a FSR of 989(8) MHz.

wavelength power (mW) trans. power (µW) ref. coe. (%) nesse421 nm 38.0(2) 28.5(2) 99.92(2) 3800(38)626 nm 10.0(2) 3.45(1) 99.96(9) 8277(62)

Another possible explanation for the low nesse is that during the evacuation watersteamed up on the mirrors. This happens especially if the evacuation occurs too fast or thevacuum chamber gets reopened without oating the chamber with argon. Since we pumpwithout any special needle valve that controls the pumping speed and had also to reopen

8Lens-Optics GmbH

65

5. Experimental Part

the chamber once, this is a good explanation. According to Lens-Optics9 the steaming eectshould get stronger for wavelengths in the UV. This agrees with the estimated nesse ofthe cavity for the light of 421 nm. Therefore this eect could be a plausible explanation forthe unexpected low nesse.

In summary, for light of 626 nm the nesse of the cavity is approximately by a factorof 4 lower than for the worst theoretical value. Since the mirrors coated for light of 626 nm,583 nm, 741 nm, and 631 nm have the highest reection at 626-nm we assume that thenesses for the other wavelengths are even worse. For the locking of the laser light thismeans, that the slope of the PDH errror signal is 4 times less than expected, becausethe slope is proportional to 1

∆ν. The transitions of 626 nm and 583 nm have linewidths of

136 kHz and 190 kHz. Since the lasers for these transitions have a small enough fee runninglinewidth the main task of the cavity is the long term stabilization. The lock of this twolasers is therefore possible. However, for the lasers of 741 nm and 631 nm as well for thelasers for Rydberg excitation the transition linewidths are below 10 kHz and therefore thelocking is, if at all possible, very challenging.

9Private communication.

66

6. Conclusion and OutlookThe main goal of this master thesis was to set up and characterize an optical cavity, whichcan be used to stabilize dierent lasers at several wavelengths. In the main experimentthese lasers will be used for slowing, cooling, and trapping of Er and Dy atoms and forRydberg excitation. To reach a high stability against temperature uctuations we choseultra low expansion glass (ULE) as material for the mirrors and the cavity spacer. Dueto technical limitations it is not possible to produce mirrors with high reectivity at allnecessary wavelengths. Therefore, we decided to built a double cavity. This means weused one ULE spacer but two dierent pairs of mirrors. One pair of mirrors is coated forthe wavelengths at 583 nm, 626 nm, 631 nm, and 741 nm, which are cooling transitions inerbium and dysprosium. The second pair of mirrors is coated for the range from 380 nm to450 nm to be able to lock lasers for Rydberg excitation.

To shield the cavity as good as possible from the environment, i. e. from acoustic andthermal noise, the cavity is placed in a vacuum chamber with a pressure on the order of2 × 10−8 mbar. For further stability, the cavity inside the vacuum chamber is surroundedby two additional heat shieldings. The temperature of the outer one of these two shieldingsis stabilized via Peltier elements. For monitoring of the temperature several thermistors aredistributed over the vacuum chamber and can be read out with an Arduino system.

The temperature where the ULE exhibits a minimal thermal expansion was determinedto be at 21.78(5) C and therefore the cavity is stabilized to this value. To determine thefree spectral range (FSR), the linewidth, and nesse of the cavity we performed threedierent measurements. First, we determined the FSR to 989(8) MHz by scanning the626-nm laser over two cavity resonances and extracted the frequency dierence of thetransmission peaks by tting a Lorentzian function. Additionally, we modulated sidebandson the laser beam with an electro-optical modulator (EOM) and used them to calibratethe frequency axis. Further, the linewidth was extracted from the t of the transmissionpeaks and was determined to 630(9) kHz, which leads to a nesse of 1569(23). As a secondmeasurement we performed a cavity ring-down and the resulting nesse was 1636(4).Finally, we extracted a linewidth of 410(20) kHz out of the PDH error signal, leading to anesse of 2213(120).

The nesse obtained with these three methods deviate by a factor of ≈ 4 form thetheoretically expected value. We come to the conclusion, that due to too fast pumping andreopening of the cavity water steamed up on the mirrors. This could have degraded thereection of the mirrors and caused therefore the lower nesse compared to the expectedvalue. Nevertheless, it was possible to lock the 626 nm laser light to the cavity.

The next step in the experiment will be the realization of the single- and double-speciesmagneto optical traps (MOTs) of Er and Dy.

67

6. Conclusion and Outlook

6.1. Er–Dy antum Mixture: Motivation and VisionSo far, quantum-mixture experiments with dipolar atoms has not yet been realized and evennever considered in theory. Our experiment would then be the rst to go in this direction.We indeed plan to combine two strongly magnetic atomic species for the rst time. Fromthe experimental realization point of view the Er–Dy mixture presents several advantages.One advantage for the combination of Er and Dy is that they share many similar properties,e. g. strength of the laser cooling transitions, polarizability, anisotropic interaction character,and similar atomic masses and melting points. Both elements give a convenient choiceof bosonic and fermionic isotopes (see Table 2.1) that gives opportunities to have threedierent types of mixtures: Bose-Bose, Fermi-Fermi and Bose-Fermi.

A rst aspect to be studied is the interspecies interaction of Er and Dy having an imbalanceon the dipolar interaction, where µ = 7µB for Er and µ = 10µB for Dy and dierenttotal angular momentum J. Both elements oer a rich spectrum of intraspecies Feshbachresonances [34, 59, 60]. The Feshbach spectroscopy of the mixture will add importantinformation for the understanding of scattering physics of dipolar gases.

For the Fermi-Fermi mixture the crossover from a Bose-Einstein Condensate (BEC) toBardeen-Cooper-Schrieer (BCS) superuid (BEC–BCS crossover) with dipole-dipole inter-action (DDI) is of large interest. The BEC-BCS crossover is important for the understandingof superconductors and the associated superuidity [82]. Depending on the interactionstrength between the atoms, given by the scattering length a, the fermions can eitherform molecules or weakly bound pairs, so called Cooper-pairs. Both the BEC and theCooper-pairs give explanation to dierent types of superuids. In many experiments theBEC–BCS crossover has been shown, e. g. in 6Li and 40K [18, 70], and the whole range fromBEC to BCS can be investigated. These experiments have in common that they work withalkali atoms having isotropic interaction. Dipolar atomic species give the possibility tostudy the crossover under the inuence of long-range and anisotropic dipolar interaction.Atoms forming a Cooper-pair have opposite momentum, because of the conservation ofmomentum [40]. In momentum space the Fermi surface for alkali atoms is spherical, dueto the isotropic interaction, whereas for dipolar atomic species the Fermi surface can bedeformed [2], which may have an impact on the Cooper-pairing mechanism. For alkaliatoms such as 6Li and 40K the experimental technique is to prepare the atoms equally intheir lowest two spin states and tune the interaction by a Feshbach resonance. However,for dipolar atoms this type of preparation is complicated by dipolar relaxation eects [14]occurring as a consequence of the long-range dipolar interaction. In a dipolar relaxationprocess the spin is ipped, since for dipol-dipol interaction the total spin is not preservedand the atoms relax into the state with lowest energy. This issue can be circumvented byusing two dierent atomic species. The idea is to prepare both species in their lowest spinstates. Again a Feshbach resonance can then be used to tune the interaction between theatoms.

Another interesting case is to investigate the behavior of the atoms without the externalmagnetic eld, that polarizes the atoms. Without the polarization, the dipoles of the atomsare not frozen anymore and will organize their orientation by them self. Spinor Bose gasesare gases with a spin degree of freedom that, similar to the BEC–BCS crossover, have been

68

6.2. Rydberg Excitation

studied in detail with alkali atoms [75]. The dipolar characteristic of Er and Dy will add newaspects. Since the dipole dipole interaction is spin dependent, this will lead to a competitionwith the spin dependent contact interaction for the local spin ordering of the spinor gas.For 87Rb in the F = 1 state the ratio (εdd) between the dipole-dipole interaction strength(add = µ0µ

2m/12π~2) and the contact interaction strength given by the scattering length a isεdd = 0.002. In comparison, for Dy, εdd is on the order of 1 for a typical scattering length [75].The large total angular momentum quantum number J of Er (J = 6) and Dy (J = 8) givesrise to the realization of spin-N systems with large N, i. e. 2J + 1 dierent states (13 for Erand 17 for Dy).

6.2. Rydberg ExcitationAn atom excited to a Rydberg state is a highly excited atom with a large principal quantumnumber n. Rydberg atoms exhibit many special properties, e. g. a large orbital radius scalingwith n2, which leads to a high polarizability, and a long lifetime proportional to n3 (forRubidium the theoretical lifetime of the 110S 1/2 state is 1.7 ms) [7, 57]. However, for Rydbergstates with a high angular momentum quantum number l, the lifetime increases furtherwith a scaling τ ∝ n5 [35]. Due to the high polarizability a large electric dipole moment canbe induced, resulting in a strong long-range interaction. Similar to a Feshbach resonance,the interaction strength is tunable [6].

Rydberg excitation using a single photon excitation scheme is technically challengingfrom the aspect of laser technology, as usually laser light in the UV is required. In orderto overcome this problem, two photon excitation schemes are commonly used in the eldof ultracold Rydberg atoms. Here, the excitation lasers are operated with a detuning withrespect to the intermediate state in order to minimize photon scattering, which eventuallyresults in a heating of the ultracold sample. However, this intermediate state detuning alsoresults in a reduction of the coupling strength between ground- and Rydberg state. Therich level structure of Er, as shown in Chap. 2, allows for various two photon excitationschemes via a narrow line intermediate state, which potentially can be utilized to increasethe coupling between ground and Rydberg state.

Further, the sub-merged shell electron conguration of Er (see Sec. 2.1) allows to exciteeither an electron in the 6s or the 4 f state. Utilizing a two photon excitation scheme, theexcitation of an 4 f electron to states with a high angular momentum and therefore, Rydbergstates up to l = 5 are reachable in a straight forward manner.

In addition, similar to alkaline earth atoms, the core of an Er Rydberg atom remainsoptically active and thus still can be addressed, e. g. for trapping, detecting or furthercooling.

69

6. Conclusion and Outlook

70

A. Appendix

A.1. Pound-Drever-Hall Error SignalThe Pound-Drever-Hall (PDH) technique [25] is a very powerful technique to frequencystabilize laser systems using a Fabry-Perot cavity as frequency reference. The PDH errorsignal is created by modulating sidebands on the laser beam and by extracting the phasedierence between the sidebands and the carrier of the laser beam after it is reected fromthe Fabry-Perot cavity. The following Appendix gives a brief summary of the derivation ofthe PDH error signal.

First, we want to concentrate on the reection of a monochromatic beam from a Fabry-Perot cavity. The incoming electrical eld can be written as

Einc = E0eiωt

while the reected beam isEref = E1eiωt

E0 and E1 are the complex amplitudes, containing the relative phase. The ratio betweenthe incoming and the reected beam leads to the reection coecient [12]

F(ω) =Einc

Eref=

r(e

iωFS R − 1

)1 − r2e

iωFS R

, (A.1)

where r is the reection coecient of the mirrors and FSR the free spectral range. If thefrequency of the laser light is exactly an integer multiple of the FSR, the phase dierencebetween the incoming and reected beam is 180° and therefore this two beams interferedestructively. If the laser frequency does not match the FSR, the phase dierence is dierentfrom 180° and the two electric eld do not cancel out each other. The phase dierenceindicates, on which side of the cavity resonance the laser frequency is on. To measure thephase of the reected beam, the incoming beam gets phase modulated, which generatessidebands on the laser beam. After the reection of the cavity the sidebands interfere withthe carrier and the phase is extracted from the beat pattern.

For the phase-modulation of the laser light and therefore, for the creation of the sidebandsa Pockels-crystal is used. An important attribute of a Pockels crystal is that the refractionindex depends linearly on the applied electric eld:

n(E) = n + aE

For the PDH technique a sinusoidal electrical eld is applied to the crystal and the electricaleld of the laser beam after the crystal can be described by

Einc = E0ei(ωt+β sin(Ωt)),

71

A. Appendix

where Ω is the frequency of the electrical eld, which is applied to the Pockels crystaland β is the parameter for the modulation depth. If β < 1, the expression of the incomingelectrical eld can be expanded using the Bessel functions [12]

Einc ≈ E0[J0(β) + 2iJ1(β) sin(Ωt)

]eiωt

where terms of higher order are neglected, because J1(β)/J0(β) ≈ 0.2 for β = 1. For β < 1 thisration becomes even smaller. Using sin(ix) = eix−e−ix

2i the electrical eld becomes

Einc = E0

[J0(β)eiωt + J1(β)ei(ω+Ω) − J1(β)ei(ω−Ω)

](A.2)

In Eq. A.2 terms with frequencies ω, ω −Ω and ω + Ω appear, where ω is the frequency ofthe carrier and ω ±Ω are the frequencies of the two sidebands. The power in the carrierand in each rst-order sideband is given by [12]

PC = J20 (β) P0 (A.3)

for the carrier andPS = J2

1 (β) P0 (A.4)

for the sidebands, where P0 is the total power of the incident beam. The distribution of thepower between the carrier and the sidebands depends on the modulation depth β. Withthe reection coecient from Eq. A.1 and the modulated incoming beam in Eq. A.2 thereected beam can be written as [12]

Eref = E0

[F(ω)J0(β)eiωt + F(ω + Ω)J1(β)ei(ω+Ω) − F(ω −Ω)J1(β)ei(ω−Ω)

](A.5)

The power measured at the photodiode is Pref = |Eref |2. In comparison to a beam at

single frequency without sidebands Eq. A.5 leads to additional terms appearing due to theinteraction of the carrier with the sidebands. The most interesting terms are the one whichoscillate at the modulation frequency Ω, because they sample the phase of the reectedcarrier:

2√

PCPSRe [F(ω)F∗(ω + Ω) − F∗(ω)F(ω −Ω)] cos(Ωt)

and

2√

PCPSIm [F(ω)F∗(ω + Ω) − F∗(ω)F(ω −Ω)] sin(Ωt).

Usually, depending on the modulation frequency only one of these two terms will survive,i. e. for low modulation frequencies F(ω)F∗(ω+Ω)−F∗(ω)F(ω−Ω) is purely real, whereasfor high frequencies it is purely imaginary. Using a mixer, which forms the product of twosignals, the terms can be extracted. The product of two sine waves is

sin(Ωt) sin(Ω0t) =12cos[(Ω −Ω0)t] + cos[(Ω + Ω0)t] (A.6)

For the case that Ω ≈ Ω0, the cosine which contains the frequency dierence becomes adc signal, which can be isolated using a low-pass lter. Thereby mixing the output signal of

72

A.1. Pound-Drever-Hall Error Signal

frequency (1/+)-6 -4 -2 0 2 4 6

norm

alize

din

tensity

-1

-0.5

0

0.5

1 r = 0.999, + = 3r = 0.9995, + = 4r = 0.9999, + = 5

(a) Overview with dierend reection coecients andmodulation frequencies.

frequency (1/+)-0.2 -0.1 0 0.1 0.2

norm

alize

din

tensity

-1

-0.5

0

0.5

1r = 0.999r = 0.9995r = 0.9999

(b) Zoom-in on the slope of the central peak.

Figure A.1.: (a) PDH error signal with 3 dierent modulation frequencies and 3 dierent reectioncoecients. (b) Slope of the PDH error signal with 3 dierent reection coecients.

the photodiode with the modulation signal and applying a low-pass lter we can extract thePDH error signal for high modulation frequencies. Since at low modulation frequencies theterm F(ω)F∗(ω+ Ω)− F∗(ω)F(ω−Ω) is purely imaginary only the cosine term is relevantand the signal need to be phase shifted to obtain the same mixing behavior. Afterwardsthe error signal can be fed to a PID controller that stabilizes the laser, e. g. by changing thelaser current and the grating angle with a piezo element for an external cavity diode laser.

In Figure A.1(a) three dierent error signals as a function of the detuning are plotted.For each signal the modulation frequency of the EOM and the reection coecient of themirrors, thus the nesse (Eq. 4.9), are changed. The change of the modulation frequencyresults in a dierent position for the wings of the error signal. A higher modulationfrequency leads to a higher capture range, i. e. the region where the error signal is unequalzero and has the correct sign for driving the laser back onto resonance. However, thedrawback of a higher modulation frequency is that the value of the error signal betweenthe wings is small and therefore the inuence on the regulation decreases. On the otherhand, the change of the reection coecient results in a displacement of the minima andmaxima, respectively. A higher reection coecient shifts the minima and maxima towardsthe center and therefore the slope around the central point increases. This central slope ofthe error signal can be expressed by [12]

D = −8√

PCPS

∆ν. (A.7)

The slope is proportional to the power of the carrier and the sidebands and to the inverseof the linewidth of the cavity. The sensitivity of the laser stabilization with the PDHtechnique to frequency uctuations is directly related to the steepness of the slope. Thesteeper the slope, the higher the inherent gain of the laser stabilization. A simple electronicamplication also amplies the present technical noise and would therefore not lead to anincreased signal-to-noise ratio. Figure A.1(b) shows the slope for three dierent reection

73

A. Appendix

coecients for the same modulation frequency. For a given linewidth of the cavity, one canmaximize the slope D by nding an optimal value for

√PCPS. The maximum value for D

can be found if PS/PC = 0.42, which corresponds to a modulation depth of β = 1.08.

A.2. Modulation-Transfer SpectroscopyThe modulation-transfer spectroscopy (MTS) is a spectroscopy technique based on twocounter-propagating beams, called pump and probe beam. The pump beam passes throughan electro-optical modulator (EOM) and sidebands are imprinted on the beam. The modula-tion is already described in Appendix A.1. After the EOM the pump beam can be representedby the carrier frequency ωC and sidebands separated by the modulation frequency ωm [63]

E = E0

[ ∞∑n=0

Jn(δ) sin (ωC + nωm) t

+

∞∑n=1

(−1)nJn(δ) sin (ωC + nωm) t − J1(δ) sin (ωC − ωm) t].

(A.8)

Jn(δ) represents the Bessel function of order n and δ is the modulation index. If themodulation index δ < 1, only the rst order sidebands have to be considered and Eq. A.8simplies to [63]

E = E0 [J0(δ) sinωCt + J1(δ) sin (ωC + ωm) t − J1(δ) sin (ωC − ωm) t] . (A.9)

Therefore, the pump beam consists of three frequency components after the EOM, i. e. ωC,ωC + ωm, and ωC − ωm. When the pump and the probe beam are overlapped in the spec-troscopy cell four-wave mixing occurs, where a fourth wave is generate by the combinationof three other waves [69]. The four-wave mixing is related to the non-linear polarization ofthe medium

PNL = χ(3)E3.

If E is the sum of three waves of three dierent frequencies ω1, ω2, and ω3, E3 leads todierent frequency combinations and the generation of a fourth wave. In the case of theMTS, two frequency components of the pump beam combine with the frequency of theprobe beam and generate a fourth frequency component that can be treated as a sidebandof the probe beam. The sidebands of the probe beam, modulated in the vapor of thespectroscopy cell, beat with the probe beam leading to [63]

UPD(∆, t) =C√

Γ2 + ω2m

×[(

L−1 − L−1/2 + L1/2 − L1)

cos(ωmt + φ

)+

(D1 − D1/2 − D−1/2 + D−1

)sin

(ωmt + φ

)],

(A.10)

whereLn =

Γ2

Γ2 + (∆ − nωm)2 (A.11)

74

A.3. Electronic Circuits

andDn =

Γ(∆ − nωm)Γ2 + (∆ − nωm)2 . (A.12)

∆ denotes the detuning of the laser frequency to the atomic resonance frequency and Γ isthe linewidth of the transition. The beat signal S (ωm) is detected with a photodiode and aphase detector mixes the signal of the photodiode with the modulation signal. Using thetrigonometric formula A.6 leads to a DC term and an oscillating term. With a low-passlter the DC term can be extracted and the spectroscopy signal

S (∆) =C√

Γ2 + ω2m

×[(

L−1 − L−1/2 + L1/2 − L1)

cos(φ)

+(D1 − D1/2 − D−1/2 + D−1

)sin

(φ)] (A.13)

obtained. It is important to note, that the modulation occur only, if the condition for thesub-Doppler resonance is fullled. Otherwise, the four-wave mixing does not transfer thesidebands onto the probe beam. This means that only atoms of the velocity class zerocontribute to the spectroscopy signal and therefore the transition is not broadened by themotion of the atoms. The resonance position of the MTS signal is always located at thecenter of the sub-Doppler resonance.

A.3. Electronic CircuitsAll used electronic circuits are designed by Dr. Manfred Mark.

Measurement BridgeThere are two measurement bridges (MBs) on one circuit, which are framed with the blackdashed line in the electronic circuit below. The MB for the resistor two with negative tem-perature coecient (NTC2) consists of has a potentiometer to set the desired temperature.The aim of the temperature PID is to regulate the resulting temperature dierence of theMB to zero. Therefore, with the MB for NTC2 we can reach the following resistance values:

RNTC = R11R8 + Rpoti

R12(A.14)

for minimum temperature and

RNTC = R11R8

R12 + Rpoti(A.15)

for maximum temperature. If instead of a NTC a PTC (positive temperature coecient) isused the minimum an maximum temperature are reversed. The resistor R8, which denesthe reachable temperature range is not predened and has to be chosen appropriately.

75

A. Appendix

For the NTC1, additionally to R16, dip switches are implemented, that can be used toadd additional resistors in series to R16. Therefore, with this measurement bridge a highertemperature range can be reached:

RNTC = R5R16 + Rpoti + Rswitch

R6(A.16)

for the minimum temperature and

RNTC = R5R16

R6 + Rpoti(A.17)

for the maximum temperature. The other elements on the circuit are either for the supplyvoltage or not used. The connection from the MB to the PID is done via a SUB-D15 connector.

Temperature PIDThe temperature PID is adapted to the measurement bridge and consists also of two indepen-dent PIDs. The circuits of both PIDs are the same, whereby for the resistors and capacitorsdierent values can be inserted to adjust the proportional, integral and dierential part.Besides the PID, there are are also other parts on the circuits such as voltage supply andmonitor output. Table A.1 gives a list of all resistors and capacitors, which we have chosenproperly.

Table A.1.: Values for the dierent components of the temperature PIDs. The designations areconsistent with the circuit below. The derivative term is not used in our case.

Component ValueRp1 1 13 kΩ

Rp1 2 100 kΩ

Rd1 –Cd1 –

C_In 1 470 µFRp2 1 11 kΩ

Rp2 2 150 kΩ

Rd2 –Cd2 –

C_In 2 100 µF

For the temperature stabilization of the cavity the derivative part of the PID is notused, because he reacts on the rate of change, which is slow for temperature. With thepotentiometer Rpot1 and Rpot2 the output current of the PID can be limited. For this theconnections P11 and P5 have to be jumpered.

76

1

1

2

2

3

3

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4

D D

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B B

A A

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a15Vin

-15Vin

GND

OUT1a

OUT1-

OUT2a

OUT2-

Vin2

Vin1

GND

GND

GND

a15V

a10V

GND

a10V Vin1

GND

GND

a15V

-15V

a10V

R810k

Vin2

12

P3

OUT2

Vset1

-15V

1V/10K

1K

R170805

1234567891011 12 13 14 15 16 17 18 19 20

P6

10 x Switc

h

5

1

2 3 4

J2

COAX-F

5

1

2 3 4

J7

COAX-F

1K

R180805

1K

R200805

1K

R190805

1K

R210805

1K

R220805

1K

R230805

1K

R240805

1K

R250805

1K

R150805

100R41206

100nF

C50805

100R101206

GND-LD

GND-LD

12

P7

NTC2-Control

12

P10

LD

GND-LD

12

P4

NTC2

12

P2

NTC1

1K

R450805

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

17

16

J1

D Connector 15

10V/K

10V/K

5k6

R160805

10k

R120805

10k

R110805

10k

R60805

10k

R50805

110RR7

1206

110R

R1

1206

12

P1

OUT1

-15V

a15Va15Vin

-15Vin

100nF

C30805

100nF

C40805

100nF

C100805

100nF

C70805

100nF

C20805

100nF

C10805

2.2uF

C131206

2.2uF

C141206GND

a15V

-15V

GND

-15V

a15V

GND

Vset12

31A

84

U4ATL052

5

67B

84

U4BTL052

100k

R31

1206

12k

R321206

82k

R331206

12k

R30

120682k

R29

1206100k

R14

1206

1k

R21206

2k2

R31206

3k3

R131206

a15V

Put resistor on socket 123

P14

Switch

GND-LD

TempPID V8a0 Messbrücke TA

Change Diode direction depending on LD Polarity

IN2

GND4 TRIM 5

OUT 6

NC 8NC1

NC3 NC 7

U1

REF01CS

1

2

3L1 NFM41P

1

2

3

L2NFM41P

-In1 RG2 RG3 aIn4

-VS 5Ref 6Vout 7aVS 8U3

AD8421

1 2 3

P5 Polarity LD

1 2 3

P8Polarity LD

GND-LD

-In1 RG2 RG3 aIn4

-VS 5Ref 6Vout 7aVS 8U2

AD8421BOT

2TO

P1

TAP 31K

R9RPot

100nF

C60805

BOT

2TO

P1

TAP 31K

R38RPot

D17V5

D2Diode

PIC101PIC102

COC1 PIC201PIC202

COC2

PIC301PIC302

COC3 PIC401PIC402

COC4

PIC501PIC502

COC5PIC601PIC602

COC6PIC701PIC702

COC7

PIC1001PIC1002

COC10PIC1301PIC1302 COC13

PIC1401PIC1402 COC14

PID101PID102 COD1

PID201PID202

COD2

PIJ101

PIJ102

PIJ103

PIJ104

PIJ105

PIJ106

PIJ107

PIJ108

PIJ109

PIJ1010

PIJ1011

PIJ1012

PIJ1013

PIJ1014

PIJ1015

PIJ1016

PIJ1017

COJ1

PIJ201

PIJ202 PIJ203 PIJ204 PIJ205

COJ2

PIJ701

PIJ702 PIJ703 PIJ704 PIJ705

COJ7

PIL101

PIL102PIL103

COL1

PIL201

PIL202PIL203

COL2

PIP101PIP102

COP1

PIP201PIP202

COP2

PIP301PIP302

COP3

PIP401PIP402

COP4

PIP501 PIP502 PIP503

COP5

PIP601PIP602PIP603PIP604PIP605PIP606PIP607PIP608PIP609PIP6010

PIP6011 PIP6012 PIP6013 PIP6014 PIP6015 PIP6016 PIP6017 PIP6018 PIP6019 PIP6020COP6

PIP701PIP702

COP7PIP801 PIP802 PIP803COP8

PIP1001PIP1002

COP10

PIP1401PIP1402PIP1403

COP14

PIR101PIR102COR1

PIR201

PIR202COR2

PIR301

PIR302COR3

PIR401 PIR402COR4

PIR501

PIR502COR5

PIR601

PIR602COR6

PIR701 PIR702COR7

PIR801

PIR802COR8

PIR901

PIR902PIR903

COR9 PIR1001 PIR1002COR10

PIR1101

PIR1102COR11

PIR1201

PIR1202COR12

PIR1301

PIR1302COR13

PIR1401PIR1402COR14

PIR1501

PIR1502COR15PIR1601

PIR1602COR16

PIR1701

PIR1702COR17PIR1801

PIR1802COR18PIR1901

PIR1902COR19PIR2001

PIR2002COR20PIR2101

PIR2102COR21PIR2201

PIR2202COR22PIR2301

PIR2302COR23PIR2401

PIR2402COR24PIR2501

PIR2502COR25

PIR2901 PIR2902COR29

PIR3001 PIR3002COR30

PIR3101 PIR3102COR31

PIR3201

PIR3202COR32

PIR3301

PIR3302COR33

PIR3801

PIR3802PIR3803

COR38

PIR4501

PIR4502COR45

PIU101PIU102

PIU103PIU104 PIU105

PIU106

PIU107PIU108

COU1

PIU201PIU202PIU203PIU204

PIU205PIU206PIU207PIU208

COU2

PIU301PIU302PIU303PIU304

PIU305PIU306PIU307PIU308

COU3

PIU401PIU402

PIU403

PIU404

PIU408 COU4A

PIU404PIU405

PIU406PIU407

PIU408 COU4B

PIC501

PIP202

PIP402PIR801

PIR1602

PIU106PIC101 PIC201 PIC601PIC701PIC1302PIL103

PIR1302

PIU102

PIU208

PIU308

PIU408

PIJ101

PIL101

PIC302 PIC402 PIC1002PIC1401PIL203

PIU205

PIU305

PIU404

PIJ103

PIL201

PIC102 PIC202PIC301 PIC401

PIC502PIC602PIC702PIC1001

PIC1301PIC1402

PIJ102

PIL102PIL202

PIR201

PIR502 PIR602

PIR1102 PIR1202

PIR3301

PIU104

PIU206

PIU306

PIJ702 PIJ703 PIJ704 PIJ705

PIP503

PIP801

PIP1001

PIP1403

PIU108PIU107PIU105

PIU103PIU101

PIR3803

PIU201

PIU403

PIR3201PIR3302

PIR3102 PIR3202PIU405PIR3101PIU401PIU402

PIR2902PIR3001PIR1401 PIR2901

PIU406

PIR1001PIU307

PIR903 PIU301

PIR901PIR1201

PIR802PIR902 PIR702PIU303PIR701

PIU302

PIR601PIR1502 PIR3801

PIR401PIU207

PIR302PIR1301

PIR202

PIR301PIR1402

PIR102PIU203

PIR101 PIU202

PIP1401

PIP6019 PIP6020 PIR2501PIR4501PIP6017 PIP6018 PIR2301PIR2401PIP6015 PIP6016PIR2101 PIR2201PIP6013 PIP6014PIR1901PIR2001 PIP6011 PIP6012PIR1701PIR1801

PIP6010PIR1601PIR1702

PIP608PIP609PIR1802PIR1902

PIP606PIP607PIR2002PIR2102

PIP604PIP605PIR2202PIR2302

PIP602PIP603PIR2402PIR2502

PIP601 PIR1501 PIR3802

PIR4502

PIP401

PIR1101

PIU304

PIP201

PIR501

PIU204

PIJ701

PIP501

PIP803

PIP1002

PIP1402

PIJ202 PIJ203 PIJ204 PIJ205PIP701

PIJ201PIP702

PIJ1017

PIJ1016

PIJ1012

PIJ108

PID102 PID202

PIP502

PID101 PID201PIP802

PIJ104

PIJ105PIP101

PIJ106

PIJ107

PIP102

PIJ109

PIJ1010PIP301

PIJ1011 PIP302

PIJ1014

PIR402

PIJ1013

PIR1002

PIJ1015

PIR3002

PIU407

1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

D D

C C

B B

A A

ExphystronixUniversity of Innsbruck

a18.05.2015 17:57:35Q:\CsIII\Documents\Drawings\MJM_Circuits_final\NeuPCB\TempPID\TempPID v8.1.SchDoc

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Technikerstraße 25

Drawn by:

6020 InnsbruckAustria

GND

GND

10Hz SallenaKey Tiefpass Invertierer/Nichtinvertierer

123

P3

INV 1 On/Off

S1INT OFF / ON

GND

GND

GNDGND

R160R1

GND

23

1 S6

Mon 1/2

A2

A4

A6

A8

A10

A12

A14

A16

A18

A20

A22

A24

A26

A28

A30

A32

C2

C4

C6

C8

C10

C12

C14

C16

C18

C20

C22

C24

C26

C28

C30

C32

P1

2x16

GND

IN2

1

OUT 3

GND

U7 7915

IN1

2

OUT 3GND

U67815

GND

GND

1 2 3 4 5 6 7 89 10 11 12 13 14 15

17 16

J1D Connector 15

2

31A

84

U1AOPA2277

5

67B

84

U1BOPA2277

GND

2

31A

84

U2AOPA2277

5

67B

84

U2BOPA2277Rp1 1

Rd1 Cd1

Poly Rp1 2

12

P11

Offset 1 On

GND

a15V

-15V

a15V-15V

-15V

-15V

-15V

a15V

a15V

a15V

GND

S3INT OFF / ON

GND

GNDGND

R340R1

GND

Vin2

5

67B

84

U5BOPA2277 2

31A

84

U5AOPA2277Rp2 1

Rd2 Cd2

Poly

Rp2 2

12

P5

Offset 2 On

GND

a15V

-15V

a15V

-15V

-15V

a15V

100uF

C17Elko

100uF

C7Elko

33uF

C8Elko

33uF

C18Elko

a15V

-15V

-15V

a15V

a15V -15VGND

OUT1a

OUT1-

OUT2a

OUT2-

OUT1a OUT1-OUT2a

OUT2-

Vin2

Vin1

Vin1

Vin1

Vin2

GND

2

31A

84 U8A

OPA22775

67B

84 U8B

OPA2277

-15V-15V

a15Va15V

GND

GND

123

P7

INV 2 On/Off

5

67B

84

U4BOPA2277

2

31A

84

U4AOPA2277

GND

-15V

-15V

a15V

a15V

12

P2

In 1

12

P6

In 2

12

P4

Out 1

12

P8

Out 2

10k

R23

120610k

R24

1206

10k

R5

120610k

R6

1206

2.2uF

C11206

1uF

C31206 100k

R9

1206

100k

R1

1206

2.2uF

C41206

1uF

C61206 100k

R27

1206

100k

R19

1206

100k

R11

1206

100k

R81206

100k

R2

1206

100k

R261206

100k

R29

1206

100k

R20

1206

10k

R14

1206

470

R171206

10k

R32

1206

470

R351206

1k

R371206

51k

R381206

4k7

R39

1206

GND

-CCW

aCW

W W10k

Rpot6Imax2

-CCW

aCW

W W10k

Rpot3Imax1

-CCW

aCW

W W5k

Rpot2Att1

-CCW

aCW

W W5k

Rpot5Att2

D2

Diode 1N4148

1k

R401206

100nF

C190805

100nF

C210805

100nF

C100805

100nF

C110805

100nF

C120805

100nF

C90805

100nF

C140805

100nF

C150805

100nF

C160805

100nF

C130805

100nF

C200805

100nF

C220805

GND

Vin1

Vin2

5

67B

84

U9BOPA2277

2

31A

84

U9AOPA2277

-15V

-15V

a15V

a15V

100k

R44

1206

100k

R45

1206

100k

R46

1206

100k

R47

1206

GND

GND

D4

Diode 1N4148

D5

Diode 1N4148

D6

Diode 1N4148

D7

Diode 1N4148

GND

100k

R481206

100k

R491206

GND

GND

VCC

100nF

C230805

1nF

C26

0805

1nF

C27

0805

aVendIn

-VendIn

GND

aVend

-Vend

100nF

C250805

10k

R41206

1k

R71206

aVend

-Vend

100nF

C20805

10k

R121206

D1

Diode 1N4148

-Vend

aVend

100uF

C5Elko

100uF

C24Elko

1 2 3

P12INT/EXT

1 2 3

P13INT/EXT

a18V -18V

aVend -Vend

aVendIn -VendIn

D3

Diode 1N4148

Vset1

GND

GND

123

9932844

P14

Display

10k

R50

1206100

R51

1206

100nF

C28

0805

GND

GND

Vset1

GND

51k

R421206

51k

R521206

100k

R41

1206

100k

R43

1206

GND

GND

optional

optional

VDD

TempPID V8a0

Logic High if Temperatureis away from Set point

IN8

NC5 NC4

GND 7

GND 3GND 2

GND 6

OUT 1U12

LM78L05ACM

1

2

3L1

NFM41P

1

2

3

L2NFM41P

21

22

P9B

Monitor

11

12

P9A ErrorOut

D8red

D9blue

100nF

C290805

100nF

C300805

a18V

-18V

1234

8765

C_In2

SW DIP-4

1234

8765

C_In1

SW DIP-4

1

2

3 45

6

7 U3OPA548T-1

1

2

3 45

6

7 U11OPA548T-1

1

23

U10A

MC74VHC132D

4

56

U10B

MC74VHC132D

12

1311

U10D

MC74VHC132D

COMMON11

NC1 2

NO1 3

COMMON24

NC2 5

NO2 6S5

SW DPDT

100nF

C310805

GND

100nF

C320805

100nF

C330805

100nF

C340805

GND

GND

- CCWaCW WW

10k

Rpot1

Offset1

- CCWaCW WW 10k

Rpot4

Offset2

PIC101PIC102

COC1

PIC201PIC202

COC2

PIC301PIC302

COC3

PIC401PIC402

COC4

PIC501PIC502

COC5

PIC601PIC602

COC6

PIC701PIC702

COC7PIC801PIC802

COC8

PIC901PIC902

COC9PIC1001PIC1002

COC10 PIC1101PIC1102

COC11 PIC1201PIC1202

COC12PIC1301PIC1302

COC13PIC1401PIC1402

COC14PIC1501PIC1502

COC15PIC1601PIC1602

COC16

PIC1701PIC1702

COC17PIC1801PIC1802

COC18

PIC1901PIC1902

COC19

PIC2001PIC2002

COC20

PIC2101PIC2102

COC21

PIC2201PIC2202

COC22

PIC2301PIC2302

COC23

PIC2401PIC2402

COC24

PIC2501PIC2502

COC25

PIC2601 PIC2602

COC26

PIC2701 PIC2702

COC27

PIC2801 PIC2802

COC28

PIC2901PIC2902

COC29PIC3001PIC3002

COC30

PIC3101PIC3102

COC31

PIC3201PIC3202

COC32

PIC3301PIC3302

COC33

PIC3401PIC3402

COC34

PIC0In101PIC0In102PIC0In103PIC0In104 PIC0In105

PIC0In106PIC0In107PIC0In108

COC0In1

PIC0In201PIC0In202PIC0In203PIC0In204 PIC0In205

PIC0In206PIC0In207PIC0In208

COC0In2

PICd101PICd102COCd1

PICd201PICd202

COCd2

PID101 PID102

COD1

PID201 PID202

COD2

PID301PID302

COD3

PID401 PID402

COD4

PID501 PID502

COD5

PID601 PID602

COD6

PID701 PID702

COD7

PID801PID802 COD8 PID901

PID902 COD9

PIJ101 PIJ102 PIJ103 PIJ104 PIJ105 PIJ106 PIJ107 PIJ108PIJ109 PIJ1010 PIJ1011 PIJ1012 PIJ1013 PIJ1014 PIJ1015

PIJ1016PIJ1017COJ1

PIL101

PIL102PIL103

COL1

PIL201

PIL202PIL203

COL2

PIP10A2

PIP10A4

PIP10A6

PIP10A8

PIP10A10

PIP10A12

PIP10A14

PIP10A16

PIP10A18

PIP10A20

PIP10A22

PIP10A24

PIP10A26

PIP10A28

PIP10A30

PIP10A32

PIP10C2

PIP10C4

PIP10C6

PIP10C8

PIP10C10

PIP10C12

PIP10C14

PIP10C16

PIP10C18

PIP10C20

PIP10C22

PIP10C24

PIP10C26

PIP10C28

PIP10C30

PIP10C32

COP1

PIP201PIP202

COP2

PIP301PIP302PIP303

COP3

PIP401PIP402

COP4

PIP501PIP502

COP5

PIP601PIP602

COP6

PIP701PIP702PIP703

COP7 PIP801PIP802

COP8

PIP9011

PIP9012

COP9A

PIP9021

PIP9022

COP9B

PIP1101PIP1102

COP11

PIP1201 PIP1202 PIP1203

COP12

PIP1301 PIP1302 PIP1303

COP13

PIP1401PIP1402PIP1403

COP14

PIR101PIR102COR1

PIR201PIR202COR2

PIR401

PIR402COR4

PIR501PIR502COR5

PIR601PIR602COR6

PIR701

PIR702COR7

PIR801

PIR802COR8

PIR901PIR902COR9

PIR1101PIR1102COR11

PIR1201

PIR1202COR12

PIR1401 PIR1402COR14

PIR1601

PIR1602COR16PIR1701

PIR1702COR17

PIR1901PIR1902COR19

PIR2001PIR2002COR20

PIR2301PIR2302COR23

PIR2401PIR2402COR24 PIR2601

PIR2602COR26

PIR2701PIR2702COR27

PIR2901PIR2902COR29 PIR3201 PIR3202

COR32

PIR3401

PIR3402COR34

PIR3501

PIR3502COR35

PIR3701

PIR3702COR37PIR3801

PIR3802COR38

PIR3901 PIR3902COR39

PIR4001

PIR4002COR40

PIR4101PIR4102COR41

PIR4201

PIR4202COR42

PIR4301PIR4302COR43

PIR4401 PIR4402COR44

PIR4501 PIR4502COR45

PIR4601 PIR4602COR46

PIR4701 PIR4702COR47

PIR4801

PIR4802COR48

PIR4901

PIR4902COR49

PIR5001 PIR5002COR50

PIR5101 PIR5102COR51

PIR5201

PIR5202COR52

PIRd101PIRd102CORd1

PIRd201PIRd202CORd2

PIRp1 101PIRp1 102CORp1 1

PIRp1 201

PIRp1 202CORp1 2

PIRp2 101PIRp2 102CORp2 1

PIRp2 201

PIRp2 202CORp2 2

PIRpot10CCWPIRpot10CW

PIRpot10WCORpot1

PIRpot20CCW

PIRpot20CWPIRpot20W

CORpot2

PIRpot30CCW

PIRpot30CWPIRpot30W

CORpot3

PIRpot40CCWPIRpot40CW

PIRpot40WCORpot4

PIRpot50CCW

PIRpot50CWPIRpot50W

CORpot5

PIRpot60CCW

PIRpot60CWPIRpot60W

CORpot6

PIS101 PIS102

COS1

PIS301 PIS302

COS3

PIS501PIS502

PIS503

PIS504PIS505

PIS506COS5

PIS601PIS602

PIS603

COS6

PIU101PIU102

PIU103

PIU104

PIU108 COU1A

PIU104PIU105

PIU106PIU107

PIU108 COU1BPIU201

PIU202

PIU203

PIU204

PIU208 COU2A

PIU204PIU205

PIU206PIU207

PIU208 COU2B

PIU301

PIU302

PIU303 PIU304

PIU305PIU306

PIU307COU3

PIU401PIU402

PIU403

PIU404

PIU408 COU4A

PIU404PIU405

PIU406PIU407

PIU408 COU4BPIU501

PIU502

PIU503

PIU504

PIU508 COU5A

PIU504PIU505

PIU506PIU507

PIU508 COU5B

PIU601

PIU602PIU603

COU6

PIU701PIU702 PIU703

COU7

PIU801PIU802

PIU803

PIU804

PIU808

COU8APIU804

PIU805

PIU806PIU807

PIU808

COU8B

PIU901PIU902

PIU903

PIU904

PIU908 COU9A

PIU904PIU905

PIU906PIU907

PIU908 COU9BPIU1001

PIU1002PIU1003

COU10A

PIU1004

PIU1005PIU1006

COU10B

PIU10011PIU10012

PIU10013

COU10D

PIU1101

PIU1102

PIU1103 PIU1104

PIU1105PIU1106

PIU1107COU11

PIU1201

PIU1202

PIU1203PIU1204PIU1205

PIU1206PIU1207

PIU1208

COU12PIC801

PIC901PIC1001 PIC1101 PIC1201 PIC2001 PIC3201

PIC3401PID302

PIJ101

PIRpot10CCW

PIRpot40CCW

PIU108PIU208

PIU408

PIU508

PIU603

PIU808

PIU908PIU1208

PIC701

PIP1201

PIS503

PIU601

PIC501PIC1901 PIC2901

PIP1202

PIU305

PIU1105

PIP10A10

PIP10A12

PIP10A14

PIP10A16

PIP10A18

PIP10A20

PIP1203

PIC1302PIC1402 PIC1502 PIC1602

PIC1802

PIC2202 PIC3302

PIJ103

PIRpot10CW

PIRpot40CW

PIU104PIU204

PIU404

PIU504

PIU703

PIU804

PIU904

PIC1702

PIP1301

PIS506 PIU702

PIC2102 PIC2402PIC3002

PIP1302

PIR402

PIR1202

PIU304

PIU1104

PIP10A22

PIP10A24

PIP10A26

PIP10A28

PIP10A30

PIP10A32

PIP1303

PIC302

PIC502

PIC602

PIC702 PIC802

PIC902PIC1002 PIC1102 PIC1202 PIC1301PIC1401 PIC1501 PIC1601

PIC1701 PIC1801

PIC1902

PIC2002

PIC2101

PIC2201

PIC2302

PIC2401

PIC2802

PIC2902PIC3001

PIC3102

PIC3202PIC3301

PIC3402

PID301

PID802 PID901

PIJ102

PIL102PIL202PIP10A4 PIP10C4

PIP201

PIP601

PIP9012

PIP9022

PIP1401

PIR1602PIR1702

PIR3402

PIR3502

PIR3701

PIR4201

PIR4801

PIR4901

PIR5102

PIR5201

PIRpot20CCW

PIRpot30CCW

PIRpot50CCW

PIRpot60CCW

PIU105

PIU203PIU205

PIU405

PIU503PIU505

PIU602PIU701

PIU903

PIU905 PIU1007

PIU1202

PIU1203PIU1206PIU1207PIU1205

PIU1204

PIU1107

PIU1006

PIU10012

PIU1003

PIU10013

PIU307

PIS602PIU805

PIS505

PIS502

PIRpot60W

PIU1101

PIRpot30W

PIU301

PIR4602

PIR4701

PIU906

PIR4402

PIR4501

PIU902

PIR4301PIR5202 PIU1001

PIU1002

PIR4101PIR4202PIU1004

PIU1005

PIR3802

PIR3901PIU801

PIR3702PIR3801

PIU802

PIR3202PIR3501 PIRpot60CWPIR2902

PIRp2 202

PIU507

PIR1902

PIR2701PIU406

PIR1402 PIR1701 PIRpot30CW

PIR1201PIU1103

PIR1102PIRp1 202PIU207

PIR401PIU303PIR102

PIR901PIU106

PIP1102

PIRpot10W

PIP1101

PIR801

PIP9021

PIU803PIU806

PIU807

PIP9011PIU10011

PIP703

PIR2702PIRpot50W

PIP702 PIR4902

PIRd202

PIRp2 102PIP701

PIR1901

PIU407

PIP502

PIRpot40W

PIP501

PIR2601

PIP303

PIR902PIRpot20W

PIP302 PIR4802

PIRd102

PIRp1 102PIP301

PIR101

PIU107

PIP10C32

PIP10C30

PIP10C28

PIP10C26

PIP10C24

PIP10C22

PIP10C20

PIP10C18

PIP10C16

PIP10C14

PIP10C12

PIP10C10

PIP10C8PIP10A8PIL203

PIS501

PIL201

PIP10A6 PIP10C6

PIL103PIS504PIL101

PIP10A2 PIP10C2

PIJ1017 PIJ1016

PIJ108

PID801 PID902PIR3902PID602

PID702

PIR4302PID601

PIR4702

PIU907

PID402

PID502

PIR4102

PID401

PIR4502

PIU901

PICd202PIRd201

PICd102PIRd101

PIC0In205PIC0In206PIC0In207PIC0In208

PIRp2 201

PIS302

PIC0In201PIC0In202PIC0In203PIC0In204

PICd201

PIRp2 101

PIS301

PIU506

PIC0In105PIC0In106PIC0In107PIC0In108

PIRp1 201

PIS102

PIC0In101PIC0In102PIC0In103PIC0In104

PICd101

PIRp1 101

PIS101

PIU206

PIC2801

PIP1402

PIR5002PIR5101

PIC2702

PID102

PIR2001

PIR3201PIU501

PIC2701

PID101

PIR2002

PIR2602PIR2901

PIU502

PIC2602

PID202

PIR201

PIR1401PIU201

PIC2601

PID201

PIR202

PIR802PIR1101

PIU202

PIC601PIR2401PIU403

PIC402

PIRpot50CWPIU401PIU402

PIC401

PIR2301PIR2402

PIC301PIR601PIU103

PIC102

PIRpot20CWPIU101PIU102

PIC101PIR501PIR602

PIC2501

PIJ104 PIJ105

PIP401PIR4001PIU306

PIC2502

PIJ106 PIJ107

PIP402

PIR1601

PIR4002

PIU302

PIC201

PIJ109 PIJ1010

PIP801PIR701PIU1106

PIC202

PIJ1011 PIJ1012

PIP802

PIR702

PIR3401

PIU1102

PIC2301 PIC3101 PIP1403

PIU10014

PIU1201

PID501

PIJ1014

PIP202PIR502

PIR4401

PIS601

PID701

PIJ1013

PIP602 PIR2302

PIR4601PIS603

PIJ1015

PIR5001

A.4. Assembly of the Vacuum Chamber

A.4. Assembly of the Vacuum ChamberIn this Appendix, we describe the main steps of the assembly of the vacuum chamber.Figure A.2 shows the base plate of the cavity with the copper plate on the Peltier elements.The Peltier elements are not glued and therefore we x the positions slightly with Kapton™tape. The copper plate is screwed to the base plate with plastic screws for better thermalisolation and to reduce the force acting on the Peltier elements. Figure A.3 shows the insideof the vacuum chamber without the shieldings. The lower ange is placed on the copperplate. We used tin foil between the Peltier elements and the lower ange to increase thethermal contact and xed the position of the Peltier elements again with Kapton™ tape.The NTCs are glued1 into small holders (see Fig. A.7), which are screwed onto the vacuumchamber at dierent positions. Figure A.4 shows the outer shielding sitting on the Pelterelements. On both sides of the shieldings windows are glued into holes (see Fig. A.5). Onthe bottom of the outer shielding, see Fig. A.6, spacers out of Teon and additional holdersfor NTCs are placed. The outer shielding is screwed onto the lower ange with vacuumcompatible M2.5 screws2. The Teon spacers reduce the thermal contact from the innershielding to the outer one. Figure. A.8 shows the cavity sitting on the V-shaped aluminumblock inside the inner shielding. To monitor the temperature of the aluminum holder twoNTCs are glued into two holes on the side of the aluminum block. The inner shielding isclosed (Fig. A.9) and the wires for to the NTC are guided through a hole in the top of theshielding to the feedthrough.

Figure A.2.: Heat sink of the vacuum chamber with the additional copper plate. Between the copperplate and the heat sink six Peltier elements are placed to temperature stabilize the copper plate.

1The NTCs are glued with Torr Seal.2Vacom, VFC-2.506-25 and VFC-2.510-25.

79

A. Appendix

Figure A.3.: Framework of the vacuum chamber with Pelter elements inside. The positions ofthe Peltier elements are xed with Kapton™and holders for the NTC are screwed on at dierentpositions.

Figure A.4.: The outer shielding sits on the Pelter elements on the lower ange.

80

A.4. Assembly of the Vacuum Chamber

Figure A.5.: Windows on the outer and inner shielding are glued in. They are coated for thenecessary linewidths, respectively.

Figure A.6.: Inside of the outer shielding. The inner one is sitting on Teon spacer to reduce thethermal contact.

81

A. Appendix

Figure A.7.: NTCs are glued into holders which can be screwed onto the vacuum chamber.

Figure A.8.: Open chamber with ULE cavity inside.

82

A.4. Assembly of the Vacuum Chamber

Figure A.9.: Both shieldings are closed and an additional temperature sensor is screwed on the top.

83

A. Appendix

84

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90

AcknowledgementFirst of all I want to thank Francesca for giving me the opportunity to do my master thesisin the RARE experiment. She supported me during the whole time, especially with thepreparation of my seminar talk and during the writing phase of my thesis. Further, I wantto thank her that she gave me the opportunity to participate at the "Enrico Fermi" SummerSchool in Varenna, which was a unique experience.

Ein großer Dank geht an alle Mitglieder des Erbium- sowie des RARE-Teams für ihreständige Hilfsbereitschaft und Geduld bei oenen Fragen. Ein ganz besonderer Dank gilt andieser Stelle Philipp, der mich von Beginn an bei der Masterarbeit unterstützt und bei Fragenimmer ein oenes Ohr gehabt hat. Weiters danken möchte ich Manfred, der mir immermit gutem Rat beiseite stand und mich sehr beim Schreiben der Masterarbeit unterstützthat. Arno möchte ich für seine Hilfsbereitschaft mit LATEX und das Lesen von Teilen derArbeit danken. Ein Dank geht auch an Emil, der mir beim Bauen des Lasers im exuredesign geholfen hat und auch sonst immer für Fragen zur Verfügung stand. Ein großerDank geht auch an die Werkstatt des IQOQI, wo Stefan, Andreas und Bernhard immerwieder große Hilfsbereitschaft gezeigt haben. Abseits der Universität möchte ich mich beider Fußballmannschaft des ASV Rian-Kuens bedanken, wo ich während meines Studiumstrotz sehr bescheidener Trainingspräsenz spielen durfte. Ein Dank geht auch an meinelangjährigen Mitbewohner Paul und Hannes, mit denen ich einige tolle Jahre zusammenwohnen durfte.

Abschließend geht der allergrößte Dank noch an meine Eltern, meine Großeltern undmeine Freundin, welche mich immer unterstützt und motiviert haben.

91

Leopold-Franzens-Universität Innsbruck

Eidesstattliche Erklärung

Ich erkläre hiermit an Eides statt durch meine eigenhändige Unterschrift, dass ich dievorliegende Arbeit selbständig verfasst und keine anderen als die angegebenen Quellenund Hilfsmittel verwendet habe. Alle Stellen, die wörtlich oder inhaltlich den angegebenenQuellen entnommen wurden, sind als solche kenntlich gemacht.

Die vorliegende Arbeit wurde bisher in gleicher oder ähnlicher Form noch nicht als Magister-/Master- /Diplomarbeit /Dissertation eingereicht.

Datum Unterschrift